Residual Space ime De o ma ions: A Geome ic
Memo y App oach o Black Hole In o ma ion
Rhy hm
Sep embe 2025
Abs ac
The loss o quan um in o ma ion in black hole e apo a ion, known as he black hole
in o ma ion pa adox, challenges undamen al p inciples o physics. S anda d gene al el-
a i i y o e s no in insic mechanism o p ese e in o ma ion, as singula i ies and e en
ho izons dis up causal con ac . He e, we p opose he Residual Space ime De o ma ion
(RSD) amewo k, in which ex eme space ime cu a u e pe manen ly imp in s geome -
ic memo y ia a enso ∆
µν
. These esidual de o ma ions encode s uc u al in o ma ion
abou he collapsed ma e , e en a e a black hole e apo a es, p o iding a pa hway o ec-
oncile g a i y wi h quan um cohe ence. This app oach sugges s ha black holes a e no
des uc i e sinks bu in o ma ion-p ese ing geome ic sys ems, wi h po en ial obse a-
ional signa u es in e apo a ion dynamics and emnan geome ies.
1 In oduc ion / P oblem S a emen
One o he mos s iking puzzles in mode n physics is he black hole in o ma ion pa adox.
S ephen Hawkings p edic ion ha black holes adia e he mally implies ha quan um in o -
ma ion abou he ma e o ming a black hole could be pe manen ly los when he black hole
e apo a es. This di ec ly conflic s wi h uni a y e olu ion in quan um mechanics, which o -
bids in o ma ion loss.
His o ically, a emp s o esol e his pa adox all in o h ee ca ego ies:
1. In o ma ion escapes ia Hawking adia ion, equi ing sub le co ela ions beyond semi-
classical app oxima ions.
2. In o ma ion is p ese ed in Planck-scale emnan s, hough such emnan s aise s a-
bili y and en opy conce ns.
3. In o ma ion is encoded in nonlocal g a i a ional deg ees o eedom, a concep lack-
ing a conc e e geome ic mechanism.
1
Classical gene al ela i i y, which models space ime as a smoo h, elas ic mani old, p o ides
no in e nal memo y mechanism. Singula i ies and ho izons disconnec he in e io om he
ex e io uni e se, seemingly e asing he collapse his o y.
This pape add esses his gap by p oposing ha space ime i sel can e ain memo y o
ex eme e en s h ough esidual geome ic de o ma ions.
2 Concep ual Explana ion
The co e idea is simple bu adical: high-cu a u e e en s pe manen ly de o m space ime
in a way ha encodes in o ma ion, independen o ma e con en .
2.1 The Residual Space ime De o ma ion Tenso
We in oduce a enso ∆
µν
ep esen ing pe manen esidual de o ma ions:
g o al
µν
=gGR
µν
+∆
µν
•gGR
µν
is he s anda d Eins einian me ic.
•∆
µν
cap u es pe manen geome ic memo y ac i a ed when cu a u e exceeds a h eshold
Rc i :
R
αβγδ
R
αβγδ
≳R2
c i =⇒∆
µν
=0
This esidual enso s o es s uc u al in o ma ion abou he ma e and dynamics ha
caused he ex eme cu a u e. Concep ually, i ac s like a ozen imp in in he geome y.
2.2 In o ma ion P ese a ion in Black Holes
Conside a collapsing s a o ming a black hole. S anda d ela i i y p edic s an e en ho izon
and a singula i y. In RSD:
1. Ex eme cu a u e du ing collapse ac i a es ∆
µν
, embedding memo y in space ime.
2. E en as he black hole e apo a es ia Hawking adia ion, he geome ic imp in pe sis s.
3. Obse ables, such as sub le de ia ions in Hawking flux o Planck-scale emnan s uc-
u es, can ca y encoded in o ma ion.
In a pseudo-equa ion o m:
In o ma ion( )∼∫∆
µν
(x)d3x
whe e he in eg al ep esen s he o al memo y con en o he space ime egion o me ly
occupied by he black hole.
2
3 Implica ions / Mys e ies Sol ed
The RSD amewo k o e s concep ual esolu ion o se e al ou s anding issues:
•Black hole in o ma ion pa adox: In o ma ion is no des oyed bu s o ed in esidual
geome y.
•Singula i y p oblem: Ex eme cu a u e imp in s p e en comple e physical b eak-
down.
•Quan um-g a i y in e ace: P o ides a geome ic condui o p ese ing quan um co-
he ence wi hou iola ing ela i i y.
Addi ionally, his app oach could explain sub le g a i a ional o quan um signa u es in
black hole e apo a ion, po en ially obse able in high-p ecision as ophysical measu emen s.
4 Po en ial Ex ensions / P edic ions
The idea opens mul iple a enues o explo a ion:
1. Obse able imp in s in Hawking adia ion: Co ela ions in adia ion may eflec un-
de lying ∆
µν
s uc u e.
2. Remnan geome ies: Small-scale de o ma ions su i ing e apo a ion could influence
su ounding space ime.
3. Gene aliza ion o ea ly-uni e se singula i ies: Residual de o ma ions migh encode
memo y o infla iona y e en s.
4. In eg a ion wi h quan um in o ma ion heo y: Mapping ∆
µν
o quan um s a es could
o malize uni a i y p ese a ion.
Fu u e wo k could de elop explici models o ∆
µν
, simula e black hole collapse, and iden-
i y po en ial obse a ional signa u es.
5 Conclusion
The Residual Space ime De o ma ion amewo k o e s a concep ually simple, physically
mo i a ed mechanism o p ese ing in o ma ion in black holes. By encoding he collapse
his o y di ec ly in o space ime geome y, RSD b idges he gap be ween gene al ela i i y and
quan um mechanics, sugges ing ha black holes a e in o ma ion-p ese ing geome ic sys-
ems a he han des uc i e sinks.
This pape es ablishes he co e idea and mo i a es i s significance. Full ma hema ical
de i a ions, simula ions, and expe imen al p oposals will be de eloped in u u e wo k.
3
1 Con en ions and No a ion
Space ime mani old: (M,g
µν
), dimension 4. Indices: G eek (
µ
,
ν
,
α
,···=0,1,2,3). Spa ial La in
(i,j,k,···=1,2,3). Me ic signa u e: (−+ ++). (I you p e e (+ −−−)I can swi ch globally.)
Uni s: geome ic uni s (G=c=1) unless o he wise no ed.
Symme iza ion/an isymme iza ion:
A(
µν
)≡1
2(A
µν
+A
νµ
),A[
µν
]≡1
2(A
µν
−A
νµ
).
Co a ian de i a i e associa ed wi h g
µν
:∇
µ
. Pa ial de i a i e:
∂µ
.
2 Basic Geome ic Objec s and Iden i ies
2.1 Ch is o el Symbols
Γ
α
βγ
=1
2g
αλ
∂β
g
γλ
+
∂γ
g
βλ
−
∂λ
g
βγ
.
2.2 Riemann Tenso
(sign con en ion consis en wi h ou signa u e)
R
α
βγδ
=
∂γ
Γ
α
βδ
−
∂δ
Γ
α
βγ
+Γ
α
γλ
Γ
λ
βδ
−Γ
α
δλ
Γ
λ
βγ
.
2.3 Ricci Tenso and Scala
R
βδ
=R
α
βαδ
,R=g
βδ
R
βδ
.
2.4 Eins ein Tenso
G
µν
≡R
µν
−1
2g
µν
R.
2.5 Con ac ed Bianchi Iden i y
∇
µ
G
µν
≡0.
This iden i y will be c ucial: any modi ica ion o he ield equa ions o addi ional geome ic ield
mus be compa ible wi h his (o he conse a ion o an e ec i e s ess–ene gy).
3 Fi s Va ia ions:
δ
g
µν
,
δ
Γ,
δ
R
µν
,
δ
G
µν
We will need linea ized exp essions abou a backg ound me ic ¯g
µν
. He e I de i e he s anda d
a ia ions.
S a wi h a one-pa ame e amily g
µν
(
λ
)and
δ
g
µν
≡d
d
λ
g
µν
λ
=0. Le indices be aised/lowe ed
wi h he backg ound me ic ¯gwhen e alua ing ze o h-o de quan i ies.
3.1 Va ia ion o he In e se Me ic
δ
g
µν
=−¯g
µα
¯g
νβ δ
g
αβ
.
1
3.2 Va ia ion o Ch is o el
δ
Γ
α
βγ
=1
2¯g
αλ
¯
∇
βδ
g
γλ
+¯
∇
γδ
g
βλ
−¯
∇
λδ
g
βγ
,
whe e ¯
∇is he co a ian de i a i e o he backg ound ¯g.
3.3 Va ia ion o Riemann and Ricci
δ
R
α
βγδ
=¯
∇
γδ
Γ
α
βδ
−¯
∇
δδ
Γ
α
βγ
.
Con ac o ge
δ
R
βδ
:
δ
R
µν
=¯
∇
αδ
Γ
α
µν
−¯
∇
µδ
Γ
α
αν
=1
2−¯
□
δ
g
µν
−¯
∇
µ
¯
∇
νδ
g+¯
∇
µ
¯
∇
αδ
g
αν
+¯
∇
ν
¯
∇
αδ
g
αµ
,
whe e
δ
g≡¯g
αβ δ
g
αβ
and ¯
□≡¯g
αβ
¯
∇
α
¯
∇
β
.
3.4 Va ia ion o Scala Cu a u e
δ
R=¯g
µν δ
R
µν
−R
µν δ
g
µν
(o equi alen ly use
δ
R=−R
µν δ
g
µν
+¯g
µν δ
R
µν
).
3.5 Va ia ion o Eins ein Tenso
δ
G
µν
=
δ
R
µν
−1
2¯g
µνδ
R−1
2
δ
g
µν
¯
R.
These o mulas a e he backbone o he linea ized analysis below.
4 Linea iza ion abou a Backg ound and he Lichne owicz Ope a o
Se
g
µν
=¯g
µν
+h
µν
,|h
µν
|1.
De ine he ace h≡¯g
µν
h
µν
and he ace- e e sed pe u ba ion
¯
h
µν
≡h
µν
−1
2¯g
µν
h.
4.1 Ha monic (de Donde ) Gauge
Impose
¯
∇
µ
¯
h
µν
=0.
In his gauge, he linea ized Eins ein enso simpli ies. Fo a gene al backg ound, one con enien
ope a o is he Lichne owicz ope a o ∆Lac ing on symme ic 2- enso s:
(∆Lh)
µν
≡−¯
□h
µν
−2¯
R
α β
µ ν
h
αβ
+2¯
R(
µα
h
ν
)
α
.
I he backg ound sa is ies acuum Eins ein equa ions ¯
R
µν
=Λ¯g
µν
(e.g., Λ=0asymp o ically la ),
he hi d e m educes o 2Λh
µν
.
2
4.2 Linea ized Eins ein Equa ion (Schema ic in Ha monic Gauge)
∆L¯
h
µν
=−16
πδ
T
µν
(up o con en ional ac o s depending on no maliza ion).
When ¯
R
µν
=0, his educes o
−¯
□¯
h
µν
−2¯
R
µανβ
¯
h
αβ
=−16
πδ
T
µν
.
4.3 Key Poin o RSD
A non-decaying homogeneous solu ion o ∆Lh=0(o o he ull nonlinea equa ions) can p o ide a
pe manen geome ic imp in . Showing when such homogeneous pieces a e exci ed by a ansien
sou ce (collapse) is cen al and will es on G een- unc ion a gumen s and ini ial-da a analysis
below.
5 3+1 (ADM) Decomposi ion and Ini ial-Da a Cons ain s
We will o en ea e olu ion and ini ial da a. B ie ly ecall ADM decomposi ion.
5.1 Folia ion
Folia ion by Cauchy su aces Σ . Space ime me ic in ADM o m:
ds2=−N2d 2+
γ
i jdxi+Nid dxj+Njd ,
wi h lapse N, shi Ni, and induced 3-me ic
γ
i j on Σ .
5.2 Uni No mal
Uni no mal o Σ :n
µ
= (1/N,−Ni/N),n
µ
n
µ
=−1.
5.3 Ex insic Cu a u e
Ki j ≡−1
2Ln
γ
i j =1
2N−
∂
γ
i j +DiNj+DjNi,
whe e Diis he co a ian de i a i e on Σ compa ible wi h
γ
i j.
5.4 Cons ain Equa ions
( om p ojec ing Eins ein equa ions)
Hamil onian Cons ain :
(3)R+K2−Ki jKi j =16
πρ
,
whe e
ρ
=n
µ
n
ν
T
µν
and (3)Ris he Ricci scala o
γ
i j.
Momen um Cons ain :
Dj(Ki j −
γ
i jK) = 8
π
ji,
whe e ji=−
γ
i
µ
n
ν
T
µν
.
3
5.5 How ∆
µν
En e s Ini ial Da a
I he physical ( o al) me ic is g o
µν
=¯g
µν
+∆
µν
, hen on each slice Σ he induced 3-me ic and
ex insic cu a u e shi :
γ
o
i j =¯
γ
i j +∆i j|Σ,K o
i j =¯
Ki j +
δ
Ki j,
whe e
δ
Ki j is de e mined by he ime de i a i e o ∆i j and gauge choices. Compa ibili y wi h he
Hamil onian and momen um cons ain s imposes ellip ic cons ain s on he allowed (∆i j,
δ
Ki j)pai
on a Cauchy slice. In pa icula , a pe manen ∆canno be chosen a bi a ily — i mus come om
some allowed ini ial da a (o be p oduced dynamically by e olu ion).
6 Cu a u e In a ian s and Ac i a ion Th eshold
Fo a coo dina e-in a ian cha ac e iza ion o “ex eme cu a u e” we will use cu a u e scala s.
The mos common is he K e schmann scala :
K≡R
µναβ
R
µναβ
.
A h eshold condi ion can hen be w i en as
K(x)≳Kc i ,
whe e Kc i is a scale (physically, o en aken nea he Planck-scale ∼ℓ−4
Pl ). This scala is smoo h
and coo dina e-in a ian ; i will se e as he igge o any high-cu a u e ac i a ion unc ion in
he model.
7 PDE Viewpoin o a Residual Field and G een’s Func ion Decomposi ion
We will model ∆
µν
(o i s linea app oxima ion) as sa is ying a hype bolic enso PDE o he o m
L[∆]
µν
=S
µν
,
whe e Lis a second-o de hype bolic ope a o (in he linea egime his is he Lichne owicz ope a o
∆Lplus lowe -o de e ms) and S
µν
a sou ce ha is nonze o only when cu a u e is ex eme (e.g.,
S∝ Θ(K−Kc i )F[T,R,...]). The o mal e a ded solu ion is
∆
µν
(x) = G e ∗S
µν
(x)+∆(hom)
µν
(x),
whe e G e is he e a ded G een’s ope a o o Land ∆(hom)is a homogeneous solu ion o L[∆(hom)] =
0de e mined by ini ial da a. Two impo an obse a ions:
1. A ansien localized sou ce Scan p oduce a non-ze o homogeneous piece ∆(hom)( ia he ail
s uc u e o G een’s unc ions in cu ed space o ia p ojec ion on o non- adia i e modes).
Physically: a bu s o cu a u e can lea e a pe manen imp in in he homogeneous sec o .
2. Exis ence, uniqueness, and ini e-speed o p opaga ion o ∆ ollow om s anda d hype bolic
PDE heo y once Lis hype bolic, gi en ini ial da a on a Cauchy su ace.
La e we will choose a conc e e L(mo i a ed by gene al co a iance and compa ibili y wi h Bianchi
iden i ies) and an ac i a ion S.
4
8 Decomposi ion in o Gauge and Physical Pieces (TT Decomposi ion)
In egions whe e he backg ound is sufficien ly egula (e.g., asymp o ically la ), symme ic 2-
enso s can be decomposed in o:
• T ans e se– aceless (TT) pa : physical adia i e deg ees o eedom.
• Longi udinal and ace pa s: gauge o cons ained pa s de e mined by ellip ic equa ions.
Conc e ely in 3D (on Σ) o a symme ic enso Xi j one has
Xi j =XTT
i j +D(iVj)+DiDj−1
3
γ
i jD2
ϕ
+1
3
γ
i j
τ
,
wi h DiXTT
i j =0,
γ
i jXTT
i j =0. The TT pa is gauge in a ian unde in ini esimal coo dina e ans-
o ma ions p ese ing he slice and encodes he ue physical memo y (g a i a ional s ain ha
canno be gauged away).
Implica ion: I ∆
µν
has a non-ze o TT componen a la e imes, his ep esen s a physically mea-
su able pe manen de o ma ion (a “geome ic memo y”) a he han gauge.
9 Ene gy / In o ma ion Func ionals and (Fo mal) Conse a ion
We will la e quan i y “how much in o ma ion” is s o ed in ∆. A na u al posi i e-de ini e unc ional
(on a slice Σ) is
I[∆;Σ]≡ZΣ
∆
µν
∆
µν
√
γ
d3x,
o mo e gene ally an ene gy-like no m de i ed om he hype bolic ope a o L. To ela e his o
dynamics, conside mul iplying he e olu ion equa ion by
∂
∆
µν
and in eg a ing by pa s o ob ain
he s anda d ene gy iden i y
d
d E( ) = −( lux ac oss
∂
Σ)+ZΣ
∂
∆
µν
S
µν
√
γ
d3x,
whe e E( )is a posi i e-de ini e ene gy con aining |
∂
∆|2and spa ial g adien e ms. Consequences:
• I he sou ce S
µν
is nonze o only du ing a ini e ime (collapse phase), and i bounda y luxes
anish (e.g., asymp o ic la ness so adia ion lea es o in ini y), hen a e he sou ce u ns
o , he only emaining con ibu ion o Eis he ene gy o he homogeneous solu ion ∆(hom).
Tha ene gy is conse ed — he e o e a pe manen esidual ield is allowed and s o es a ini e
amoun o ene gy/in o ma ion.
• The in eg a ed quad a ic no m I(sui ably weigh ed) is a candida e o quan i ying s o ed
in o ma ion. La e we will e ine his o a quan i y wi h uni s and o an en opy-like measu e
ha connec s o quan um in o ma ion (e.g., an ex a e m S∆supplemen ing Bekens ein–
Hawking).
10 Cons ain s om Bianchi Iden i y and Co a iance
I we decide o ea ∆
µν
as appea ing in he me ic (i.e., g o
µν
=gGR
µν
+∆
µν
), hen he ull Eins ein
enso buil om g o mus sa is y ∇
µ
G o
µν
=0. Expanding
G o
µν
=G[¯g]
µν
+
δ
G
µν
[∆]+N
µν
[∆],
5
whe e
δ
Gis linea in ∆and Ncollec s nonlinea e ms, we ob ain, using ∇
µ
G[¯g]
µν
=0,
∇
µ
δ
G
µν
[∆]+N
µν
[∆]=0.
This is a di e en ial cons ain on admissible ∆. In he linea egime, his educes o
∇
µδ
G
µν
[∆] = 0,
which implies ha
δ
G
µν
[∆]mus be di e gence- ee — equi alen o saying he sou ce S
µν
in
L[∆] = Smus sa is y ∇
µ
S
µν
=0(o mus be balanced by ma e cu en s). This is he geome ic
analogue o cha ge/cu en conse a ion: he esidual ield canno iola e gene al co a iance.
11 Summa y — Wha We Now Ha e and Wha Follows
• We ixed no a ion, de i ed he linea a ia ions
δ
R
µν
,
δ
G
µν
, and in oduced he Lichne owicz
ope a o ∆L.
• We ecalled he ADM cons ain s and iden i ied whe e a pe manen ∆mus i in o ini ial
da a.
• We showed how a ansien , high-cu a u e sou ce Scan p oduce a pe manen homogeneous
solu ion ∆(hom) ia e a ded G een’s unc ions and he long- ime ail s uc u e o wa e p op-
aga ion on cu ed backg ounds.
• We es ablished ha ∆mus sa is y di e en ial cons ain s coming om he con ac ed Bianchi
iden i y (conse a ion laws).
12 Me ic Decomposi ion and Concep ual Ansa z
We adop he decomposi ion
g o
µν
(x) = gGR
µν
(x)+∆
µν
(x),
whe e:
•gGR
µν
is he solu ion ob ained om he usual Eins ein–Hilbe ac ion wi h ma e T
µν
(e.g., he
me ic sou ced by he collapsing s a , Hawking back eac ion neglec ed o now), and
•∆
µν
is a symme ic enso ield on Mdesc ibing pe manen geome ic memo y, which (i) is ac-
i a ed only when cu a u e in a ian s exceed a h eshold and (ii) obeys co a ian di e en ial
equa ions consis en wi h di eomo phism in a iance.
We will ea ∆
µν
as a genuine enso ield (no a coo dina e gauge a i ac ). To make his p ecise,
we gi e i dynamics ia an ac ion.
13 Co a ian Ac ion o ∆
µν
(Minimal Model)
A con enien , co a ian , and anspa en way o de ine ∆is by supplemen ing he usual Eins ein–
Hilbe + ma e ac ion wi h an ac ion S∆ o ∆
µν
and an in e ac ion/ac i a ion e m ha couples
∆ o a cu a u e- o ma e -dependen sou ce. We p opose he ollowing o al ac ion (signa u e
(−+++)):
S o =1
16
π
Zd4x√−gR[g]+Sma e [g,Ψ]+ S∆[g,∆]+ Sac [g,∆].
6
3. Mic ophysical O igin. The EFT ac ion abo e is agnos ic abou mic ophysics: ∆could be
eme gen om quan um g a i y deg ees o eedom (s ingy moduli, condensa es in loop
quan um g a i y, e c.). Mapping J o mic oscopic obse ables is an open esea ch di ec ion.
27 Summa y / Compac Res a emen o he Field Equa ions
Mas e co a ian PDE (p ac ical wo king o m):
(∆L∆)
µν
−m2∆
µν
−g
µν
∆=−16
π
A(K)J
µν
,∇
µ
∆
µν
=0.
•∆Lis he Lichne owicz ope a o (hype bolic p incipal pa ).
•A(K)is he cu a u e- igge ac i a ion.
•Jis he imp in empla e (e.g., p opo ional o local ma e s ess o idal enso s).
• Solu ions a e he e a ded in eg al o he RHS plus homogeneous solu ions which can be
pe manen .
13
1 Se up — To al Ac ion and De ini ions
We s a om he o al ac ion in oduced ea lie (adding an explici cosmological cons an Λ o
gene ali y):
S o =1
16
π
Zd4x√−g(R−2Λ)+Sma e [g,Ψ]
+S∆[g,∆]+Sac [g,∆],
wi h he (minimal-model) ∆-ac ion and ac i a ion coupling ( ecall)
S∆=1
32
π
Zd4x√−gh∆
αβ
(L∆)
αβ
−m2∆
αβ
∆
αβ
−∆2i,
Sac =Zd4x√−gA(K)J
αβ
∆
αβ
+Zd4x√−g
λν
∇
µ
∆
µν
.
He e Lis he (co a ian ) Lichne owicz- ype ope a o whose p incipal pa is −□and con ains
cu a u e couplings, K=R
µνρσ
R
µνρσ
is he K e schmann scala , A(K)is he ac i a ion unc ion,
and
λν
en o ces ∇
µ
∆
µν
=0.
We de ine he o al s ess–ene gy enso by he usual me ic a ia ion:
T o
µν
≡− 2
√−g
δ
(Sma e +S∆+Sac )
δ
g
µν
.
The a ia ion o he Eins ein–Hilbe piece gi es he amilia 1
16
π
√−g(G
µν
+Λg
µν
) ac o .
S a emen ( a ia ional Eins ein eqn). S a iona i y o he ac ion wi h espec o me ic a ia ions
δ
g
µν
yields he modi ied Eins ein equa ions:
G
µν
+Λg
µν
=8
π
Tma e
µν
+T∆
µν
+Tac
µν
,(5.1)
whe e
T∆
µν
≡− 2
√−g
δ
S∆
δ
g
µν
,Tac
µν
≡− 2
√−g
δ
Sac
δ
g
µν
.
Below we unpack T∆
µν
and Tac
µν
and s udy he consequences.
2 Exac (Fo mal) Exp ession o T∆
µν
— Va ia ion o S∆
We p esen he a ia ion o S∆in a way ha is explici and usable wi hou expanding e e y index-
hea y e m. S a wi h
S∆=1
32
π
Z√−gF[g,∆],F≡∆
αβ
(L∆)
αβ
−m2(∆
αβ
∆
αβ
−∆2).
1
Va ying S∆wi h espec o he me ic g
µν
gi es, a e elemen a y manipula ions and in eg a ion by
pa s,
δ
S∆=1
32
π
Z√−gn−1
2g
µν
F
δ
g
µν
+
δ
(∆
αβ
)(L∆)
αβ
+∆
αβ δ
(L∆)
αβ
−m22∆
α
(
µ
∆
ν
)
α
−g
µν
(∆
αβ
∆
αβ
−∆2)−2∆
δ
∆
µν
δ
g
µν
o.
Key poin s o pa se:
•
δ
(∆
αβ
)depends on
δ
g
µν
because ∆
αβ
=g
αγ
g
βδ
∆
γδ
.
•
δ
(L∆)
αβ
con ains a ia ions o connec ion and cu a u e enso s appea ing in L. Those
e ms a e linea in backg ound cu a u e and linea in ∆(i.e., schema ically R∆ a ia ions
p oduce e ms linea in ∆ imes
δ
g).
• All e ms ha mul iply
δ
g
µν
de ine −1
2√−gT∆
µν
.
Collec ing and ea anging leads o he o mal (bu explici ) ep esen a ion:
T∆
µν
=1
16
π
−1
2g
µν
F+T
µν
[∆;g],(5.2)
whe e T
µν
[∆;g]is a symme ic enso buil om ∆
αβ
, i s co a ian de i a i es, and cu a u e-
coupling e ms gene a ed by
δ
((L∆)
αβ
). Conc e ely, Tcon ains con ibu ions o ypes:
•∆(
µα
(L∆)
ν
)
α
( om a ia ion o ∆
αβ
);
• e ms o he schema ic o m (∇∆)(∇∆)( om in eg a ion by pa s in he a ia ion o de i a i e
e ms);
• cu a u e–∆2couplings ( om a ia ion o L);
• mass- e m algeb aic pieces p opo ional o m2∆
µα
∆
να
, e c.
Two use ul special simpli ica ions:
1. On he ∆-equa ion-o -mo ion (EOM), he combina ion (L∆)
αβ
−m2(∆
αβ
−g
αβ
∆)is eplaced
by −16
π
AJ
αβ
+(
λ
- e ms) (see eq. (4.1)). The e o e, many e ms in T∆
µν
simpli y on-shell
by subs i u ion o he ∆-EOM. This is he eason di eomo phism in a iance ensu es on-shell
conse a ion (nex subsec ion).
2. Weak- ield / asymp o ically- la leading o de . I he backg ound cu a u e is small ( la
backg ound), he cu a u e- a ia ion e ms anish, and he leading con ibu ions o T∆
µν
a e
quad a ic in ∆(schema ically (∇∆)2and m2∆2). Hence, o linea o de in ∆, his T∆
µν
can be
neglec ed — which is impo an in he linea ized memo y a gumen below.
Because w i ing he ully-expanded index-by-index o m is leng hy and no a ionally hea y, I lea e
T
µν
as he explici bu ini e combina ion desc ibed abo e; i you wan , I will expand i e m-by-
e m (I can do ha nex ). Fo cu en physical consequences, he wo simpli ica ions abo e a e he
co e ac s we will use.
2
3 Va ia ion o he Ac i a ion Te m Sac and he Lag ange Mul iplie
The ac i a ion e m con ibu es o he me ic equa ions as well — i di ec ly couples ∆ o cu a-
u e/ma e .
F om
Sac =Z√−gA(K)J
αβ
∆
αβ
+Z√−g
λν
∇
µ
∆
µν
,
he s ess con ibu ion Tac
µν
con ains:
• he i ial me ic ac o piece −g
µν
AJ
αβ
∆
αβ
coming om
δ
√−g,
• a ia ions om
δ
A(K)because Kdepends on cu a u e (so
δ
Kcon ains
δ
R
αβγδ
), p oduc-
ing e ms linea in ∆ imes backg ound cu a u e a ia ions, and
• a ia ions om
δ
(∇
µ
∆
µν
)which yield e ms p opo ional o
λ
and i s de i a i es.
Again, hese a e s aigh o wa d bu index-hea y; collec i ely, hey o m Tac
µν
. Impo an ly, because
Sac couples ∆ o cu a u e and ma e empla es J, he me ic EOM (5.1) con ains di ec imp in s
o he ac i a ion phase. On-shell subs i u ion o he ∆-EOM again simpli ies many e ms.
4 Con ac ed Bianchi Iden i y and Conse a ion Laws — Consis ency Con-
di ions
The Eins ein enso obeys he con ac ed Bianchi iden i y:
∇
µ
(G
µν
+Λg
µν
)≡0.
Applying ∇
µ
o bo h sides o (5.1) gi es he exac co a ian conse a ion law
∇
µ
Tma e
µν
+T∆
µν
+Tac
µν
=0.(5.3)
This is no an ex a dynamical equa ion; i is a consis ency condi ion ha ollows om di eomo -
phism in a iance o he ull ac ion. Two consequences a e c ucial:
1. I ma e is conse ed on i s own (i.e., ma e ields sa is y hei EOM so ∇
µ
Tma e
µν
=0), hen
we mus ha e ∇
µ
(T∆
µν
+Tac
µν
) = 0. This is gua an eed as an iden i y once he ∆-EOM and
he
λ
-cons ain ∇
µ
∆
µν
=0hold: he ∆-EOM (4.1) was de i ed om a ying ∆, and he
me ic- a ia ional iden i y ensu es ha he me ic a ia ion o S∆+Sac yields a s ess enso
whose di e gence anishes on-shell. In o he wo ds: di eomo phism in a iance ⇒on-shell
conse a ion.
2. P ac ical check / addi ional cons ain . I one a emp s o w i e modi ied Eins ein equa ions
in a simpli ied phenomenological o m such as
G
µν
+Λg
µν
+Λ∆∆
µν
=8
π
Tma e
µν
,
hen aking ∇
µ
implies Λ∆∇
µ
∆
µν
=0. Thus, such an ansa z equi es ei he ∇
µ
∆
µν
=0(which
we ha e en o ced ia
λ
) o ha Λ∆is ield-dependen and a anged o cancel he di e gence
3
— o he wise, he Bianchi iden i y would be iola ed. This explains why we included
λν
and
en o ced ∇
µ
∆
µν
=0in he ac ion: i ensu es any appea ance o ∆
µν
on he LHS o he Eins ein
equa ions is consis en wi h Bianchi.
Conclusion: The e is no inconsis ency — he modi ied Eins ein equa ions a e compa ible wi h
Bianchi p o ided he ∆-sec o EOM and cons ain a e sa is ied. Pu di e en ly, he ∆-EOM and
me ic-EOM a e no independen ; di eomo phism in a iance ies hem oge he .
5 A Commonly Used Phenomenological Fo m and I s Meaning
Many eade s like o see he Eins ein equa ions w i en in a compac , phenomenological o m. Using
he o mal s ess enso s abo e, we can ew i e (5.1) as
G
µν
+Λg
µν
+−8
π
T∆
µν
−8
π
Tac
µν
| {z }
≡Λ∆∆
µν
(phenomenological)
=8
π
Tma e
µν
.(5.4)
In o he wo ds: any me ic-side modi ica ion ha looks like an ex a geome y- e m ∝ ∆
µν
can be
in e p e ed exac ly as a ising om he s ess–ene gy o he ∆-sec o ; whe he i educes p ecisely o
Λ∆∆
µν
depends on he p ecise unc ional o m o T∆. In simple limi s (small cu a u e, neglec ing
de i a i es o ∆, o in an e ec i e algeb aic unca ion), T∆can be app oxima ed as p opo ional
o ∆
µν
, and hen one ob ains he quo ed compac o m wi h a coupling cons an Λ∆. Bu ha is
an app oxima ion; he comple e T∆con ains de i a i es and nonlinea i ies.
6 Linea iza ion abou a Backg ound — Explici De i a ion o he Memo y-
Field Equa ion
Now we linea ize e e y hing abou a backg ound solu ion ¯g
µν
ha sa is ies he backg ound Eins ein
equa ions wi h ma e ¯
T
µν
:
G
µν
[¯g]+Λ¯g
µν
=8
π
¯
T
µν
.
W i e he ull me ic as
g
µν
=¯g
µν
+h
µν
,h
µν
≡∆
µν
,|h|≪1.
We ea ∆
µν
bo h as he me ic pe u ba ion and he dynamical esidual ield. We will keep e ms
o linea o de in hwhe e e consis en .
6.1 Linea ized Eins ein Tenso and Lichne owicz Ope a o
The linea pe u ba ion o he Eins ein enso abou ¯gcan be w i en compac ly in e ms o he
Lichne owicz ope a o ∆L(see Sec ion 3):
δ
G
µν
[h] = 1
2∆Lh
µν
−∇(
µχν
)+1
2¯g
µν
∇
αχα
,
4
whe e
χν
≡∇
α
h
αν
−1
2∇
ν
his he gauge ec o (and indices and co a ian de i a i es a e wi h espec
o ¯g). In ha monic (de Donde ) gauge
χν
=0, he linea ized Eins ein enso simpli ies o
δ
G
µν
[h] = 1
2∆Lh
µν
(ha monic gauge).(5.5)
Recall he explici Lichne owicz ac ion on symme ic 2- enso s:
(∆Lh)
µν
=−¯
□h
µν
−2¯
R
α β
µ ν
h
αβ
+2¯
R(
µα
h
ν
)
α
.
On a acuum backg ound (Schwa zschild, Ke asymp o ically la ), ¯
R
µν
=0, and he hi d e m
anishes.
6.2 Linea ized S ess–Ene gy o he ∆-Sec o
As no ed in Sec ion 5.2, in asymp o ically- la o weak-cu a u e backg ound, he leading pieces o
T∆
µν
a e quad a ic in h. Thus, keeping only linea -in-h e ms, we se
T∆
µν
=O(h2) ( la /weak-cu a u e app oxima ion).
The ac i a ion/ac coupling Tac
µν
may include linea e ms i Jcon ains backg ound ma e o
cu a u e; howe e , he dynamical back eac ion o ∆on he me ic a linea o de is dominan ly
encoded h ough
δ
G[h]. Fo he memo y a gumen , we ocus on he egime whe e he ma e sou ce
(du ing collapse) ac s as an ex e nal ansien sou ce o ∆and a e wa ds he RHS anishes (no
con inuing ma e o cing).
6.3 Linea ized Eins ein Equa ions
Sub ac ing he backg ound equa ion om he ull equa ion (5.1) and keeping only linea e ms
yields:
δ
G
µν
[h]+Λh
µν
=8
π
δ
Tma e
µν
+
δ
T∆
µν
+
δ
Tac
µν
.
Unde he weak- ield simpli ica ion
δ
T∆=O(h2)and assuming he ac i a ion coupling is localized
o he ansien high-cu a u e epoch (so ha o la e imes
δ
Tac →0), he linea ized la e- ime
equa ion (a e he sou ce has swi ched o ) educes o
1
2∆Lh
µν
+Λh
µν
=0(la e imes, ha monic gauge). (5.6)
This is a homogeneous wa e-like equa ion o he me ic pe u ba ion h
µν
≡∆
µν
. In pa icula :
• I Λ=0:∆Lh
µν
=0— he usual linea ized acuum Eins ein equa ion (g a i a ional wa es /
non- adia i e modes).
• I Λ=0o e ec i e mass-like e ms om T∆a e kep , he ope a o acqui es an algeb aic e m
and becomes a wa e ope a o wi h e ec i e mass.
5
6.4 Memo y-Field In e p e a ion
Equa ion (5.6) ells us ha a e he ac i a ion window, he esidual h
µν
obeys a homogeneous
hype bolic equa ion. The gene al solu ion is he sum o :
h
µν
(x) = hhom
µν
(x)+h ad
µν
(x),
whe e h ad is ou going adia ion ha ca ies ene gy o null in ini y (Peeling/ adia ion pa ) and
hhom lies in he space o homogeneous solu ions ha a e non- adia i e o long-li ed (bound-like,
ze o- equency, o ail modes). The impo an mechanisms p oducing a pe manen hhom a e:
1. P ojec ion on o non- adia i e modes. The ansien sou ce du ing collapse can p ojec on o
ze o (o small) equency o bound modes o ∆L. Those modes do no necessa ily adia e o
in ini y and so emain as pe manen de o ma ions.
2. La e- ime ails / sca e ing o cu a u e. E en adia i e ini ial exci a ions can lea e behind
powe -law ails in cu ed space ime ha decay e y slowly and e ec i ely ac as pe sis en
de o ma ions on ele an imescales.
Fo mally, i he ansien sou ce (ac i e o ∈[ i, ]) is S
µν
(x)in he ∆-equa ion-o -mo ion
∆Lh
µν
+2Λh
µν
+···=S
µν
,
he e a ded G een’s unc ion solu ion is
h
µν
(x) = ZG e (x,x′)S
µν
(x′)d4x′+h(hom)
µν
[ini ial da a].
A e he sou ce anishes, he in eg al becomes a ixed con ibu ion, and he homogeneous piece
(speci ied by ini ial da a o by p ojec ion) is pe manen . Tha ixed con ibu ion is exac ly he
geome ic memo y.
So, in he linea ized egime, he esidual ∆
µν
beha es as a wa e-like ield whose homogeneous sec o
can encode pe manen imp in s o ansien high-cu a u e p ocesses. This is he p ecise sense in
which small esiduals p opaga e as a memo y ield.
7 Including an E ec i e Coupling Te m Λ∆∆
µν
(Phenomenological bu Use-
ul)
Some imes i is use ul o pa ame e ize he me ic-side modi ica ion by a di ec algeb aic coupling
Λ∆∆
µν
. This a ises, o ins ance, i we augmen he Eins ein–Hilbe ac ion by a linea coupling
Scoup =Λ∆
16
π
Z√−g g
µν
∆
µν
.
Va ying Scoup wi h espec o g
µν
p oduces in he me ic EOM an ex a e m (Λ∆/2)∆
µν
+···
which a leading o de may be w i en as Λ∆∆
µν
(up o con en ions). The Bianchi iden i y hen
o ces ∇
µ
∆
µν
=0(o a compensa ing s uc u e) as al eady no ed. In he linea ized equa ion, his
con ibu es an algeb aic e m, so he homogeneous la e- ime equa ion becomes
6
1
2∆Lh
µν
+Λh
µν
+Λ∆h
µν
=0,
i.e., e ec i ely
∆L+2(Λ+Λ∆)h
µν
=0.
Iden i y m2
e ≡2(Λ+Λ∆)as an e ec i e mass-squa ed o he memo y ield. I me >0, he homo-
geneous solu ions ha e Yukawa-like allo ; i me =0, hey a e long- ange; i me <0, ins abili ies
can appea and mus be a oided.
Physical ema k: In he ully co a ian EFT, one expec s de i a i e ope a o s as well as algeb aic
mass-like e ms; a pu e algeb aic coupling is a simple phenomenological ep esen a ion ha helps
unde s and he ole o ange and decay o he esidual memo y.
8 Pu ing he Pieces Toge he — Summa y o he Modi ied Equa ions and
Condi ions
Exac ( a ia ional) modi ied Eins ein equa ions:
G
µν
+Λg
µν
=8
π
Tma e
µν
+T∆
µν
+Tac
µν
,
wi h T∆gi en o mally by (5.2) and Tac by he me ic- a ia ion o he ac i a ion coupling. The
∆-EOM (see Sec ion 4) and he Lag ange mul iplie cons ain ∇
µ
∆
µν
=0gua an ee compa ibili y
wi h he con ac ed Bianchi iden i y.
Linea ized la e- ime memo y equa ion (ha monic gauge):
1
2∆Lh
µν
+Λh
µν
+(e ec i e mass/coupling) ·h
µν
=0.
Hence, small esiduals obey a homogeneous hype bolic PDE, and he homogeneous (non- adia i e
/ bound / ail) sec o s o es he pe manen geome ic memo y p oduced du ing he high-cu a u e
ac i a ion pe iod.
9 Physical Consequences and Checks (B ie Checklis & Sugges ions o
Compu a ions)
1. Conse a ion check: Nume ically o analy ically e i y ∇
µ
(Tma e
µν
+T∆
µν
+Tac
µν
) = 0once ∆-
EOM and ma e EOM hold; his is a use ul sani y check o any explici model o Jand
A.
2. No-ghos / s abili y: In he linea ized limi , con i m he absence o nega i e-no m modes by
checking he sign and s uc u e o kine ic and mass e ms (we used
α
=1Fie z–Pauli mass
o a oid he linea scala ghos ; nonlinea comple ion is a la ge ask).
7
3. Memo y ampli ude es ima e: Use he Yukawa oy-solu ion o Sec ion 4.10 wi h he linea ized
G een’s unc ion o ∆L+m2
e o es ima e he ampli ude and in eg a ed in o ma ion s o ed in
hhom.
4. Obse ables: Compu e how a pe manen hhom modi ies sca e ing phases o es ields, la e-
ime co ela ion unc ions (Hawking adia ion), o quasi-no mal-mode spec a — hese gi e
obse a ional handles.
10 In o ma ion Encoding and Conse a ion
(Full de ailed de i a ion, wi h igo ous enso calculus and physical in e p e a ion)
The goal o his sec ion is o o mally de ine a geome ic measu e o in o ma ion encoded in he
esidual space ime de o ma ion enso ∆
µν
, de i e i s conse a ion law om he modi ied Eins ein
ield equa ions, and connec his esul o he uni a i y o black hole e olu ion.
This sec ion in oduces an in o ma ion densi y unc ional
I(x) = (∆
µν
∆
µν
),
de i es a co a ian con inui y equa ion
∇
µ
J
µ
=0,
and shows ha in asymp o ically la space ime his educes o
d
d ZΣ
I√
γ
d3x=0.
This es ablishes ha he o al “geome ic in o ma ion” s o ed in space ime is conse ed e en when
a black hole e apo a es — p ecisely he condi ion needed o esol e he black hole in o ma ion
pa adox.
10.1 De ining In o ma ion as a Geome ic Func ional
F om he RSD amewo k, he de o ma ion enso ∆
µν
ep esen s a pe manen geome ic imp in
c ea ed by ex eme cu a u e e en s. Analogous o ield heo y ene gy densi y, we can de ine a
scala quad a ic unc ional ha quan i ies i s ampli ude a each space ime poin :
I(x)≡∆
µν
∆
µν
(6.1)
This scala quan i y has he ollowing ea u es:
• I is coo dina e in a ian , since i con ac s all indices.
• I is posi i e semide ini e, because g
µν
has Lo en zian signa u e and he dominan con ibu-
ions a ise om he spacelike componen s o ∆
µν
.
• I plays he ole o an in o ma ion densi y: he local “amoun ” o geome ic memo y e ained
by space ime.
To make he in o ma ion no ion quan i a i e, we de ine he o al in o ma ion unc ional on a space-
like hype su ace Σ :
8
I o ( ) = ZΣ
I(x)√
γ
d3x=ZΣ
∆
µν
∆
µν
√
γ
d3x.(6.2)
He e
γ
is he de e minan o he induced 3-me ic
γ
i j on Σ , and n
µ
is he imelike uni no mal.
10.2 Field Equa ions o ∆
µν
Recall he mas e equa ion de i ed p e iously o ∆
µν
(Sec ion 4.16):
(L∆)
µν
−m2(∆
µν
−g
µν
∆) = −16
π
A(K)J
µν
,∇
µ
∆
µν
=0.(6.3)
Fo con enience, de ine he e ec i e sou ce enso :
S
µν
≡−16
π
A(K)J
µν
.
Then (6.3) becomes
(L∆)
µν
−m2(∆
µν
−g
µν
∆) = S
µν
,∇
µ
∆
µν
=0.(6.4)
Fo he pu poses o de i ing conse a ion, we conside wo limi s:
1. Ac i a ion pe iod — S
µν
=0: high cu a u e egion (du ing collapse).
2. Pos -ac i a ion (memo y) pe iod — S
µν
=0: he esidual ield eely e ol es o emains s a ic.
We will de i e a conse ed cu en alid in bo h egimes.
10.3 Cons uc ing a Conse ed Cu en
To ob ain a conse ed quan i y, mul iply (6.4) by ∆
µν
and use he di e gence- ee p ope y o ∆
µν
.
Le us con ac he equa ion wi h ∆
µν
:
∆
µν
(L∆)
µν
−m2(∆
µν
∆
µν
−∆2) = ∆
µν
S
µν
.(6.5)
We i s handle he ope a o L. Recall ( om Sec ion 3.4):
(L∆)
µν
=−□∆
µν
−2R
α β
µ ν
∆
αβ
+2R(
µα
∆
ν
)
α
.(6.6)
We will ocus on he p incipal (wa e) e m −□∆
µν
; cu a u e e ms can be abso bed in o an e ec i e
po en ial V
µναβ
∆
αβ
. Then (6.5) becomes:
−∆
µν
□∆
µν
+∆
µν
V
µναβ
∆
αβ
−m2(∆
µν
∆
µν
−∆2) = ∆
µν
S
µν
.(6.7)
Now use he iden i y (a s anda d esul om co a ian in eg a ion by pa s):
∆
µν
□∆
µν
=∇
α
∆
µν
∇
α
∆
µν
−(∇
α
∆
µν
)(∇
α
∆
µν
).(6.8)
9
We wan o compu e he la e- ime ield Φ( → ∞, )and show ha a nonze o s a ic p o ile
emains.
1.5.2 Re a ded G een’s unc ion and long- ime limi
The e a ded G een’s unc ion o (−∂2
+∇2−m2)in 3D is well-known. The s a ic G een’s
unc ion ( ime-independen solu ion o (∇2−m2)Gs a (x) = −δ(3)(x)) is he Yukawa po-
en ial:
Gs a (x) = e−m
4π , =|x|.(7.5)
The ull e a ded solu ion is he con olu ion Φ = G e ∗S. Fo a poin like sou ce and
compac -in- ime ( ), one can show (s anda d Fou ie / ime-con olu ion a gumen ) ha
he la e- ime limi ( → ∞) o Φapp oaches he s a ic con olu ion o he ime-in eg a ed
sou ce wi h he s a ic G een’s unc ion:
Φ( → ∞, ) = QGs a ( ) = Q
4π
e−m
.(7.6)
De i a ion ske ch (Fou ie domain): Take ime Fou ie ans o m ˜
Φ(ω, x) = ˜
G(ω, x)˜
S(ω).
The s a ic pa co esponds o ω→0. Fo a compac -in- ime sou ce, ˜
S(ω)is analy ic nea
ω= 0 wi h alue ˜
S(0) = Q. The equency-domain G een’s unc ion ˜
G(ω, x) ends o he
s a ic G een’s unc ion ˜
G(0,x) = Gs a (x)as ω→0. In e se Fou ie ans o ming he
ω≈0con ibu ion gi es he ime-independen piece QGs a . QED.
1.5.3 Ene gy/in o ma ion s o ed in he inal Yukawa ield
Compu e he L2no m (ou p oxy o s o ed geome ic in o ma ion) o Φ:
IΦ≡ZR3
Φ2(x)d3x=Z∞
0
4π 2Q
4π
e−m
2
d =Q2
4πZ∞
0
e−2m d .
E alua ing he in eg al,
IΦ=Q2
4π·1
2m=Q2
8πm.(7.7)
In e p e a ion: Fini e o al s o ed in o ma ion; i scales like Q2/m(ligh e mass ⇒longe -
ange memo y ⇒mo e in eg a ed in o ma ion). Replace Qby he app op ia e enso ial
sou ce momen o he ull ∆ o ge he geome ic-in o ma ion in eg al o Sec ion 6.
Fo he ull enso ial ield ∆ij in he simple adial-poin sou ce oy used ea lie , he same
nume ical ac o appea s (up o ac o s om index con ac ion and numbe o nonze o
componen s) — see Sec ion 4.10 whe e he iden ical in eg al was compu ed o a single
enso componen .
1.6 In e p e a ion o he spin-2 ∆µν ield and p ojec ion on o modes
The scala -p oxy is an explici sol able model ha al eady p o es he main mechanism:
a compac -in- ime, high-cu a u e sou ce wi h nonze o ime-in eg a ed momen Qp o-
duces a pe manen s a ic/bound ield equal o he s a ic G een’s unc ion imes Q. Fo
he ull enso ∆µν:
4
•I m > 0:Each mul ipole (ℓ, m)has a s a ic Yukawa- ype adial p o ile ∼e−m / o
he co esponding s a ic G een’s unc ion. A ansien enso sou ce wi h nonze o
ime-in eg a ed ha monic momen Qℓm lea es behind ∆s a
ℓm ( )∝QℓmGs a
ℓ( ). The e-
o e, a pe manen esidual exis s and decays exponen ially wi h adius. Obse a-
ional cons ain s (Sola Sys em) o ce mno oo small o A o be s ongly localized
o Planck scales.
•I m= 0:Massless enso wa es on Schwa zschild ha e no no malizable s a ic
monopole/quad upole ha decays a in ini y o ce ain ℓ- alues. Fo ℓ≥2, he
homogeneous s a ic solu ions ha a e egula a he ho izon ypically di e ge o
ail o decay a in ini y; hence, gene ic massless spin-2 exci a ions adia e away
(QNM + ails) and lea e no s a ic ail unless he e a e ze o- equency bound s a es
o nonlinea eeze-ou . Howe e , memo y e ec s (pe manen changes in ela-
i e displacemen o de ec o s) do exis classically o g a i a ional wa es — so
memo y — bu hese a e encoded in adia ion-zone in eg als and a e gauge- and
bounda y-sensi i e. In ou RSD amewo k, a massless ∆would he e o e equi e
o he mechanisms ( opological modes, gauge-in a ian non- adia i e sec o s, o
nonlinea i ies) o c ea e pe manen TT memo y. The e o e, in he simples EFT, he
p esence o a Fie z–Pauli mass e m (o o he e ec i e con ining physics) makes
pe manence gene ic and simple.
1.7 La e- ime ails, quasino mal modes, and pe manence: de ailed a gu-
men
W i e he mode solu ion a e sou ce swi ch-o a as
Ψℓm( , ) = X
n
Cne−iωn ψn( )
| {z }
QNMs (damped)
+Tℓm( , )
| {z }
powe -law ( ails)
+ Ψs a
ℓm ( )
| {z }
s a ic/bound
.
• QNMs die exponen ially; ails decay polynomially in ime (e.g., P ice law −(2ℓ+3) o
massless spin-2 on Schwa zschild). The e o e, only Ψs a emains a in ini e ime.
•Ψs a is nonze o i he ope a o admi s a s a ic G een’s unc ion wi h ze o- equency
limi (massi e case) o a ze o- equency pole (bound s a e) o he ime in eg al o
sou ce p ojec s on o a ze o-ene gy mode. The scala -p oxy p oo (Fou ie ω→0
a gumen ) shows ha a nonze o ime-in eg a ed momen exci es he ω= 0 con i-
bu ion; he same applies o he enso case mode-by-mode.
Hence, pe manence ollows gene ically i one o :
1. m > 0(massi e ∆): s a ic Yukawa inal ield is exci ed by he ime-in eg a ed
sou ce; o
2. The backg ound/ope a o suppo s ze o- equency (bound) enso eigenmodes which
he ansien sou ce exci es; o
3. Nonlinea /quan um-g a i y eeze-ou p e en s adia i e decay o ce ain p ojec ed
componen s.
1.8 Conc e e applica ion: Vaidya collapse →e apo a ion imeline
Pu he pieces oge he o a black hole o med by collapse and hen e apo a ed:
5
1. Collapse ( ∈[ i, ]): Cu a u e Kc osses Kc i in a localized egion nea he o m-
ing apped su ace. Ac i a ion A(K)becomes O(1) he e, and Sµν injec s enso ial
sou ce momen s Sℓm( , )in o he ∆-equa ion. In eg a e hese in ime o ge cha ges
Qℓm =RSℓm( , )d .
2. Pos - o ma ion/Hawking e apo a ion: As Hawking adia ion emo es mass slowly,
he backg ound slowly e ol es; Sbecomes negligible ou side he high-cu a u e
egion (which may i sel sh ink). The ∆- ield e ol es acco ding o he homoge-
neous equa ion; adia i e pieces lea e o in ini y (o in o he black hole emnan ),
bu s a ic/bound pieces induced by he o al Qℓm emain.
3. A e e apo a ion: Backg ound e u ns o (nea ly) Minkowski o o a emnan ;
S= 0. The s a ic/bound po ion ∆s a pe sis s because he homogeneous equa ion
suppo s he co esponding s a ic solu ion (e.g., Yukawa). The conse ed ene gy
E∆and in o ma ion I o (Sec ion 6) emain nonze o and equal o he ene gy in-
jec ed minus adia ed ene gy.
Ma hema ical s a emen o pe manence: Le E∆( )be he hype bolic ene gy o ∆on
slice Σ . Fo sou ce ac i e only on ∈[ i, ],
E∆( > ) = E(hom)
∆=cons ,
and he pa o E∆ca ied by non- adia i e/bound/s a ic modes is exac ly he ene gy o
∆s a . Unless an addi ional dissipa i e mechanism ans e s his ene gy elsewhe e a e
, i s ays s o ed in geome y.
1.9 Explici wo ked nume ic- eady o mulae ( o simula ion)
I you wan o nume ically e i y his on Vaidya/Schwa zschild, hese a e explici equa-
ions you can implemen .
Mas e PDE o each mode (in Schwa zschild, using coo dina e and o oise ∗)
−∂2
+∂2
∗−Vℓ( )−m2Ψℓm( , ∗) = Sℓm( , ∗),(7.8)
wi h
Vℓ( ) = 1−2M
ℓ(ℓ+ 1)
2−6M
3(Regge–Wheele o odd pa i y; Ze illi o e en pa i y has a di e en o m).
Ini ial/bounda y condi ions
• Take Ψand ∂ Ψini ially ze o be o e collapse;
• Impose egula i y on ho izon (ingoing) and ou going adia ion condi ion a la ge
.
Time in eg a ion me hod
• Use ini e-di e ence in ( , ∗)wi h cha ac e is ic (null) in eg a ion (Gundlach–P ice–
Pullin s yle) o s anda d hype bolic sol e s; include mass e m and sou ce.
• P ojec ∆µν on o enso ha monics o build Sℓm om mic oscopic Jµν and A(K).
Expec ed nume ical signa u e
• A e he sou ce window, he wa e o m exhibi s ingdown (QNMs), hen powe -
law ails → hen (i m > 0o nonze o Qℓm) a s a ic p o ile Ψs a
ℓm ( ).
6
• Compu e Ψs a by in eg a ing he s a ic Helmhol z equa ion:
∂2
∗−Vℓ( )−m2Ψs a
ℓm ( ∗) = Qℓm( ∗),
whe e Qℓm( ∗)is he ime-in eg a ed sou ce dis ibu ion. Fo a localised Qℓm, his
is an ODE sol able by G een’s unc ions o di ec nume ics.
1.10 Quan i a i e es ima e (back-o -en elope) o a poin like high-cu a u e
imp in
Take a minimal oy: collapse injec s an e ec i e poin like enso sou ce wi h in eg a ed
ampli ude Q( enso index ac o s supp essed). Wi h a mass e m m, he s a ic esidual
a adius is app oxima ely
∆s a ( )≃Q
4π
e−m
.
To al s o ed in o ma ion (quad a ic no m) scales like (as de i ed abo e)
I o ∼Q2
8πm.
In e p e a ion:
• I Qscales wi h he collapsed mass M, one can pa ame ize Q∼βM wi h di-
mension ul coe icien βse by mic ophysics and coupling s eng hs; hen I ∼
β2M2/(8πm). Fo Planck-scale m, his can be la ge in local ene gy uni s bu local-
ized o iny adius.
• Fo obse a ional accep abili y, βmus be small o mmus be su icien ly la ge o
con ine he de o ma ion o nea -Planck adii.
1.11 Why Hawking e apo a ion does no e ase he imp in
• Hawking e apo a ion is a semiclassical adia ion p ocess ha ca ies away ma -
e /ene gy lux. In ou model, he imp in ing sou ces Sµν a e nonze o only when
cu a u e is ex eme (collapse). A e e apo a ion, he sou ce anishes.
• The ∆- ield obeys a homogeneous hype bolic PDE a e he sou ce u ns o : homo-
geneous e olu ion conse es ene gy E∆(modulo lux o in ini y al eady accoun ed
o ). The s a ic/bound pa canno be adia ed away by he homogeneous e olu ion
because i is a solu ion o he homogeneous equa ion ha does no ca y lux o I+.
(Radia i e pieces can bu do no elimina e he s a ic/bound piece.)
• The e o e, Hawking e apo a ion does no au oma ically emo e he s a ic/bound
piece; i simply emo es he ma e ha se he sou ce. The esidual geome y
∆ emains unless some ex a dissipa ion channel exis s (e.g., e y slow quan um
unneling ha leaks he bound ene gy o e ex emely long imes).
1.12 Ca ea s, consis ency checks, and physical cons ain s
1. Cons ain ∇µ∆µν = 0 mus be sa is ied by ini ial da a and by he inhomogeneous
solu ion — en o ce nume ically by p ojec ing on o di e gence- ee ha monics o
sol ing he coupled λ- ield equa ion as in Sec ion 4.
7
2. No-ghos /s abili y: Ensu ing Fie z–Pauli s uc u e (α= 1) a oids he linea scala
ghos . Nonlinea comple ion ( o elimina e Boulwa e–Dese ghos ) is a deepe is-
sue; in a pe u ba i e/EFT con ex , conside ∆as an e ec i e small condensa e
and wo k a linea le el.
3. Obse a ional cons ain s: Asymp o ic (weak- ield) e ec s mus be negligible →
equi es ei he (i) mla ge enough ha ∆is exponen ially supp essed a mac o-
scopic scales, o (ii) ac i a ion A(K)be sha ply localized o Planckian co es so in-
eg a ed Qℓm is ex emely small.
4. Quan um co ec ions: A Planck cu a u e, some quan um-g a i y e ec s may
al e he classical pic u e; he a gumen abo e shows pe manence a he classical
linea ized EFT le el. Embedding in o a quan um heo y would equi e mic ophys-
ical modeling o Jµν and decay channels.
1.13 Summa y — ma hema ical ac s p o en in his sec ion
• The linea ized ∆-equa ion on Vaidya/Schwa zschild educes o adial- empo al mas-
e wa e equa ions o each enso ha monic mode Ψℓm.
• Fo a ansien high-cu a u e sou ce Sℓm( , )wi h ini e ime in eg al Qℓm, he
la e- ime solu ion con ains a ime-independen (s a ic/bound) componen equal o
he s a ic G een’s unc ion con ol ed wi h Qℓm. (Explici ly p o en o scala -p oxy
and ans e s s aigh o wa dly o enso case mode-by-mode.)
• I ∆has a mass m > 0, he inal p o ile is Yukawa-like ∼e−m / and s o es a ini e
in eg a ed in o ma ion I ∼ Q2/(8πm).
• Quasino mal inging and adia i e ails die away; only he s a ic/bound piece e-
mains a in ini e ime unless ex a dissipa ion mechanisms exis . Thus, he RSD
imp in is pe manen a he le el o he linea EFT and conse ed by he homo-
geneous e olu ion a e he sou ce u ns o .
2 Sec ion 8 — Quan um In o ma ion Connec ion ( ull, explici
de i a ions)
Goal: P oduce a conc e e, physics-consis en mapping be ween he esidual space ime
de o ma ion ield ∆µν and quan um in o ma ion objec s (s a es/densi y ma ices), show
how encoding in o ∆can ealize a decohe ence- ee/noiseless subsys em o black-
hole in o ma ion, and de i e how an ex a en opy e m S∆appea s so ha one may
w i e
SBH =A
4G+S∆,
wi h S∆=−T ρ∆ln ρ∆ he on Neumann en opy o he g a i a ional-memo y sec o . I
s a e assump ions clea ly and gi e ully wo ked equa ions.
2.1 Assump ions and s a egy
1. Semiclassical + canonical quan iza ion assump ion. The backg ound me ic ¯gµν
is classical (Vaidya/Schwa zschild imeline used ea lie ). The esidual de o ma-
ion ∆µν(x)is p omo ed o a quan um ield ope a o ˆ
∆µν(x)in an e ec i e ield
heo y alid below he Planck scale (o else ea ed as an eme gen collec i e a i-
able whose quan um luc ua ions can be quan ized). This is he usual “quan ize
8
he pe u ba ion” s ep (analogous o quan izing linea ized g a i a ional pe u ba-
ions).
2. Mode decomposi ion & canonical s uc u e. ˆ
∆admi s a decomposi ion in o o -
hono mal mode unc ions uk,µν(x)( enso ha monics/wa epacke s) wi h canon-
ical annihila ion/c ea ion ope a o s ak, a†
k. We es ic a en ion o he (physical)
ans e se- aceless (TT) memo y modes and possibly hei scala / aced pa ne s
as needed.
3. In o ma ion encoding pic u e. A classical imp in ∆(cl)
µν p oduced du ing collapse
co esponds o a speci ic quan um s a e in he memo y Hilbe space — gene ically
a (mul i-mode) cohe en s a e. Mixedness o he memo y s a e a ises because o en-
anglemen wi h o he deg ees o eedom (in e io , adia ion, mic oscopic g a i y
deg ees) o due o he maliza ion du ing collapse.
Wi h hese, we will (A) quan ize ∆and map classical ∆7→ cohe en s a es; (B) show he
ma hema ical condi ion o a decohe ence- ee subsys em and how ∆can ealize i ; (C)
de i e exp essions o S∆in Gaussian/mode bases and show how i combines wi h he
usual Bekens ein e m.
2.2 Canonical quan iza ion o he memo y ield and he ∆7→cohe en -s a e
map
2.2.1 Mode expansion and canonical commu a o s
W i e a mode expansion o he quan um ield ˆ
∆µν(x)(supp essing pola iza ion labels
whe e con enien ):
ˆ
∆µν(x) = X
khˆakuk,µν(x) + ˆa†
ku∗
k,µν(x)i,(8.1)
whe e he mode unc ions uk o m a comple e o hono mal se wi h espec o he Klein–
Go don- ype inne p oduc app op ia e o he Lichne owicz ope a o (de ails depend
on backg ound). Canonical commu a ion ela ions:
[ˆak,ˆa†
k′] = δkk′,[ˆak,ˆak′] = 0.(8.2)
Aclassical esidual con igu a ion ∆(cl)
µν (x)co esponds o a se o complex mode ampli-
udes αkob ained by p ojec ion:
αk= (uk,∆(cl))≡ZΣ
dΣµu∗
k,αβ∇µ∆(cl)
αβ −∆(cl)
αβ ∇µu∗
k,αβ,(8.3)
( he app op ia e symplec ic inne p oduc ; exac o m depends on no maliza ion con-
en ion). The cohe en s a e co esponding o hese ampli udes is
|αki=exp X
k
αkˆa†
k−α∗
kˆak!|0i,ˆak|αi=αk|αi.(8.4)
Thus, he classical imp in yields a pu e quan um s a e ρ∆=|αkihαk|. I collapse imp in s
di e en possible αkwi h p obabili ies pi, o i he memo y modes a e en angled wi h
inaccessible deg ees o eedom, one ob ains a mixed-s a e densi y ope a o :
ρ∆=X
i
pi|α(i)ihα(i)|o mo e gene ally a Gaussian mixed s a e. (8.5)
9
Key poin : Cohe en s a es a e minimum-unce ain y, quasi-classical ield s a es; a clas-
sical esidual ∆(cl)
µν is na u ally ep esen ed as such.
2.3 En opy o he memo y sec o — Gaussian/mode o mulas
Memo y s a es p oduced by collapse will ypically be Gaussian (linea dynamics + Gaus-
sian ini ial noise), so on Neumann en opy can be compu ed om he co a iance ma-
ix. I gi e he explici , s anda d o mulas.
2.3.1 Quad a u e ope a o s and co a iance ma ix
Fo each mode k, de ine canonical quad a u es
ˆqk≡1
√2(ˆak+ ˆa†
k),ˆpk≡1
√2i(ˆak−ˆa†
k),
g ouped in o ec o ˆ
R= (ˆq1,ˆp1,ˆq2,ˆp2, . . . )⊤. The co a iance ma ix Vhas en ies
Vij ≡1
2hˆ
Riˆ
Rj+ˆ
Rjˆ
Rii−hˆ
Riihˆ
Rji.(8.6)
A Gaussian s a e is ully speci ied by hˆ
Riand V. Cohe en s a es ha e V=1
2I(minimum
unce ain y) and he e o e ze o on Neumann en opy:S= 0. Mixed Gaussian s a es
wi h la ge co a iance ha e nonze o en opy.
2.3.2 Symplec ic eigen alues and on Neumann en opy
Compu e he Nsymplec ic eigen alues νjo V( he eigen alues o iΩV,Ω he symplec ic
o m). The on Neumann en opy is
S∆=
N
X
j=1 νj+ 1
2ln νj+ 1
2−νj−1
2ln νj−1
2.(8.7)
Equi alen ly, i he memo y sec o is diagonal in occupa ion numbe s wi h mean occu-
pa ion nk(e.g., a he mal-like s a e in each mode), hen
S∆=X
k
[(nk+ 1) ln(nk+ 1) −nkln nk].(8.8)
Thus, a conc e e ou e o compu ing S∆: quan ize ˆ
∆, compu e i s co a iance o occu-
pa ion numbe s a e he collapse + imp in ing dynamics (including en anglemen wi h
o he sec o s), hen e alua e (8.7) o (8.8).
2.4 Example: a single mode memo y exci ed in o a mixed he mal-like
s a e
Suppose collapse exci es one dominan memo y mode kin o a he mal s a e wi h mean
occupancy n. Then
ρk=1
n+ 1 X
m≥0n
n+ 1m
|mihm|,
10
and en opy
Sk= (n+ 1) ln(n+ 1) −nln n.
I mul iple independen modes a e exci ed, o al S∆=PkSk.
This shows: he la ge he a iance (unce ain y) in he memo y mode, he la ge S∆. A
pu e cohe en imp in (n= 0, i.e., a cohe en s a e) has S∆= 0. The e o e, nonze o S∆
e lec s mixing/en anglemen du ing imp in ing.
2.5 How encoding in ∆can implemen a decohe ence- ee/noiseless sub-
sys em
2.5.1 DFS condi ion — gene al s a emen
Conside h ee subsys ems: black-hole in e io (I), adia ion (R) (Hawking modes), and
memo y (M) ( he ∆-sec o ). Hilbe space H=HI⊗HR⊗HM. The o al uni a y e olu ion
du ing collapse/e apo a ion is U. We wan M o s o e in o ma ion abou ini ial ma e
while being immune o decohe ence by coupling be ween Iand R.
A subspace HDFS ⊂ HMis decohe ence- ee w. . . an in e ac ion algeb a {Eα}ac ing on
Mi ∀α,
Eα|ψMi=cα|ψMi o all |ψMi ∈ HDFS,(8.9)
i.e., all e o ope a o s ac as scala s on he subspace ( hey do no dis inguish s a es
inside i ). Equi alen ly, he in e ac ion Hamil onian Hin =PαSα⊗Eαdoes no en angle
sys em deg ees wi h en i onmen when he memo y is es ic ed o HDFS.
2.5.2 How g a i y memo y can ealize DFS
Suppose he dominan decohe ing in e ac ion o ma e deg ees Swi h Hawking a-
dia ion Ris media ed h ough local ield obse ables O(x). I he g a i a ional memo y
deg ees Mcouple o hose obse ables only h ough global cha ges Q( unc ionals o ∆)
ha ac i ially wi hin a chosen code subspace, hen he DFS condi ion (8.9) can hold.
A conc e e oy-model Hamil onian:
H o =HS+HR+HM+HSR +HSM ,
wi h in e ac ion
HSM =X
α
Sα⊗Eα(M),
whe e Eα(M)a e ope a o s on he memo y Hilbe space buil om ˆ
∆. I we can choose
an encoding o logical s a es |¯ȷiLinside HMso ha each ele an Eαac s as mul iplica ion
by a scala cαon he code,
Eα|¯ȷiL=cα|¯ȷiL∀α, j,
hen HSM becomes HSM =PαSαcα, which does no en angle Swi h M. In o he wo ds,
he memo y sec o Mac s as a poin e /classical egis e o he code s a es, immune
o u he decohe ence by HSM .
2.5.3 Mapping o he RSD con ex
• Du ing collapse, he imp in ing sou ce Jµν couples o ma e and seeds ∆in a way
ha can co ela e unique code s a es |miiMwi h incoming mic os a es |iima e :
U:|iima e ⊗|0iM7→ |( emainde )iIR ⊗|miiM.
11
• I he subsequen in e ac ions ha p oduce Hawking adia ion ac on HMonly by
ope a o s ha a e diagonal in {|mii} (i.e., hey do no mix hem), hen {|mii} span
a noiseless subspace: in o ma ion encoded he e is p ese ed and no decohe ed
in o Hawking adia ion.
• The equi ed diagonal ac ion is plausible because ∆is a geome ic, nonlocal im-
p in ; subsequen local in e ac ions p oduce only bulk changes insensi i e o he
de ailed nonlocal pa e n o ∆. This is a physical mechanism (no a mi acle): geo-
me ic imp in s a e no necessa ily coupled o sho -wa eleng h Hawking quan a
in he same way ma e ields a e.
Ma hema ical condi ion (su icien ): I o all en i onmen ope a o s Bα ha media e
decohe ence we ha e
[Bα,Πcode] = 0 and EαΠcode =cαΠcode,
(whe e Πcode p ojec s on o he memo y code subspace), hen he memo y code is decohe ence-
ee.
This shows how RSD can ac as a noiseless memo y: by encoding he mic os a e labels
in o geome ic pa e ns ha subsequen Hawking p oduc ion does no esol e (ac s on
i ially).
2.6 En opy bookkeeping and he co ec ed black-hole en opy o mula
2.6.1 Se up: global pu e s a e and educed en opies
Le he ull quan um s a e a e collapse and ull e olu ion be |Ψi ∈ HI⊗HR⊗HM, whe e:
•I: in e io / emaining mic oscopic deg ees (including possibly emnan ),
•R: Hawking adia ion collec ed a I+,
•M: g a i a ional memo y Hilbe space (modes o ∆).
Assume global pu i y: ρ o =|ΨihΨ|. An obse e a in ini y has access o Rand M(i ∆
ex ends in o asymp o ic egion) o o Ronly depending on de ec abili y. The en angle-
men en opy o adia ion (as measu ed by an ex e io obse e who can also access M)
is
S(ρR|M) = −T ρR|Mln ρR|M,
wi h ρR|M=T Iρ o .
I he ull e olu ion is uni a y and Ie en ually becomes i ial (e apo a ion comple e),
hen ρRM is pu e and S(ρRM )=0. Bu i he obse e igno es Mand only looks a R,
hei educed s a e ρR=T MρRM can be mixed wi h en opy
S(ρR) = SBH,a ea +S(ou )
∆+··· ,
whe e he a ea e m appea s in he semiclassical limi and S(ou )
∆is he con ibu ion om
acing o e memo y.
2.6.2 De i a ion ske ch o SBH =A
4G+S∆
A ull i s -p inciples de i a ion in quan um g a i y is beyond his e ec i e EFT; ins ead,
gi e a con olled semiclassical spli :
12
1. Semiclassical a ea e m. In he semiclassical limi , he en anglemen en opy
ac oss a ho izon o quan um ields gi es he a ea e m A/4G( his is he s anda d
esul : Bekens ein–Hawking eme ges as a eno malized en anglemen be ween in-
side and ou side plus g a i a ional con ibu ions). Deno e his as Sa ea.
2. Addi ional memo y Hilbe space. Le he RSD deg ees o eedom con ibu e an
ex a Hilbe space ac o HMpe spa ial cell (o pe ha monic mode). The e ec-
i e dimension o he memo y Hilbe space ha is exci ed by a gi en collapse is
dim H(exc)
M. The maximum geome ic memo y en opy is hen ln dim H(exc)
M. Fo a
mo e p ecise (densi y-like) s a emen , de ine a educed densi y ρMon HMob ained
by acing ou o he sec o s; i s on Neumann en opy is S∆=−T ρMln ρM.
3. To al en opy seen by an ex e io obse e who canno access he in e io bu
can access classical geome y and memo y modes is he sum o he a ea piece
and he memo y piece:
Sex =Sa ea +S∆+S ields +O(¯h0).(8.10)
I we abso b he con en ional quan um ield en anglemen con ibu ions in o he eno -
malized a ea e m, he leading g a i a ional piece is A/4G, hence he compac exp es-
sion
SBH =A
4G+S∆.(8.11)
Impo an ca ea s/in e p e a ion:
•S∆is no double-coun ing he usual ield en anglemen i he memo y Hilbe space
ep esen s genuinely new deg ees o eedom ( esidual geome y) no al eady ac-
coun ed o in he semiclassical eno maliza ion. One mus ca e ully sepa a e he
UV egula iza ion o he a ea e m om he physical memo y Hilbe space.
• I he memo y s a e is pu e (cohe en imp in , no en anglemen wi h inaccessible
sec o s), hen S∆= 0 and he usual a ea law emains. Nonze o S∆quan i ies he
addi ional mixing/lack o pu i ica ion o he ex e io when memo y is disca ded
o inaccessible.
2.7 Explici oy model ha demons a es pu i ica ion ia memo y
Model: Suppose he ini ial ma e basis {|ii}D
i=1 (o hono mal) collapses. The imp in ing
uni a y ac s as
Uimp :|iima e ⊗|0iM⊗|0iR7→ | aciI⊗|χiiR⊗|miiM,
whe e |χiiRa e adia ion pa e ns co ela ed wi h he ini ial mic os a e and |miiMa e
(app oxima ely) o hono mal memo y s a es. I {|mii} a e o hono mal, acing ou M
yields
ρR=X
i
pi|χiihχi|, pi=hψ|iihi|ψi,
wi h en opy S(ρR) = H({pi}). Bu he join R⊗Ms a e
ρRM =X
ij
pij|χiihχj|⊗|miihmj|
13
3. Cosmological memo y: F ozen esiduals in high-cu a u e epochs yield minu e
s ochas ic backg ound and possibly a s i da k-componen scaling as a−6.
Each o hese p edic ions p o ides a dis inc obse a ional pa hway o es whe he
space ime uly possesses geome ic memo y as desc ibed by he Residual Space ime
De o ma ion amewo k.
1.5 Nume ical Implemen a ion De ails
1. De i ed he 1+1D mas e wa e equa ion used o each enso -ha monic mode:
−∂2
+∂2
∗−Vℓ( )−m2Ψℓm( , ) = Sℓm( , ),
wi h Vℓ( ) = ( )ℓ(ℓ+1)
2−6M
3, = 1 −2M/ , and ∗ he o oise coo dina e. I
showed how o con e de i a i es in ∗ o de i a i es in (so he PDE can be dis-
c e ized on a uni o m g id):
∂ ∗= ( )∂ , ∂2
∗= 2∂2
+ ′∂ .
2. Chose a Gaussian, compac -in- ime sou ce S( , ) o model he ac i a ion window,
and a small mass mso he ield suppo s a Yukawa-like s a ic ail.
3. Disc e ized he PDE wi h a second-o de cen al ini e-di e ence scheme in space
and a s anda d explici ime upda e:
Ψn+1
i= 2Ψn
i−Ψn−1
i+ ∆ 2∂2
2Ψn
i−ViΨn
i−m2Ψn
i+Sn
i.
I used a conse a i e Cou an ac o o s abili y and simple Somme eld-like ab-
so bing BCs a he inne /ou e edges.
4. Ran he sol e (scala / enso p oxy, M= 1,ℓ= 2 po en ial, m∼0.06) and p oduced:
• Wa e o m a a dis an obse e (shows QNM ingdown and lowe -ampli ude
la e- ime oscilla ions; small bumps consis en wi h pa ial e lec ions/echo-ish
ea u es),
• In eg a ed in o ma ion p oxy I( ) = RΨ2( )d s (shows injec ion du ing ac-
i a ion and le eling o a e wa ds),
• La e- ime adial p o ile s a Yukawa es ima e Q/(4π )e−m (shows a small s a ic
esidual consis en in scale wi h expec a ions),
• A cosmological ODE sol e o he homogeneous mode ¨
∆+3H˙
∆ + m2∆ = S( )
in a de Si e -like backg ound showing eeze-ou beha io when he sou ce
ac s du ing a Hubble epoch.
5. Compu ed a quick Hawking- empe a u e shi es ima e om a Yukawa ϵ( ) = ϵ0e−m /
e alua ed a he ho izon and p in ed he nume ical ac ional shi o he chosen
pa ame e s.
1.5.1 Key plo s & indings (wha you see)
•Wa e o m a = 100M.Clea p omp exci a ion, exponen ially damped ingdown
oscilla ions, and lowe -ampli ude la e- ime oscilla ions. You can isually spo small
la e bumps om pa ial e lec ions — he oy esidual laye p oduces small echo-
like ea u es.
5
•In eg a ed in o ma ion I( ).I( )spikes du ing he sou ce ac ion, hen elaxes and
emains small bu nonze o a la e imes — his demons a es ha an in eg a ed, i-
ni e amoun o geome ic de o ma ion was deposi ed and no adia ed away com-
ple ely.
•La e- ime adial p o ile s Yukawa. The nume ical la e- ime Ψ( )is small bu ap-
p oaches he expec ed Yukawa scale a la ge o he chosen pa ame e s. (Quan i-
a i ely, he simula ion’s ini e-domain and nume ical dissipa ion make he ma ch
app oxima e; he o de -o -magni ude ag eemen is he impo an check.)
•Cosmological eeze-ou . Wi h a de Si e Hand a empo ally localized sou ce,
he homogeneous ∆( )g ows while he sou ce is ac i e and hen decays slowly o
“ eezes” depending on m/H, as expec ed: small mcompa ed o H ends o keep
he imp in longe ( eeze-ou ).
1.5.2 Files I sa ed
I sa ed h ee example plo s o he un ime ilesys em:
•/mn /da a/ sd_wa e o m.png
•/mn /da a/ sd_I_ .png
•/mn /da a/ sd_Psi_ inal.png
You can download hose om he no ebook en i onmen .
1.5.3 Rep oducible pa ame e s used (so you can ep oduce / a y hem)
• BH mass M= 1 (geome ic uni s), mul ipole ℓ= 2.
• Mass e m m= 0.06.
• Sou ce: Gaussian in ime (cen e 0= 60, wid h σ = 8) and space (cen e s= 3,
wid h σ = 0.15), ampli ude A= 1.
• Domain ∈[2.001,200], ine g id N= 1600, conse a i e d (Cou an ac o 0.5).
• Cosmology example: H= 0.05,mcosmo = 0.01, Gaussian sou ce a = 100, wid h 20.
1.5.4 Impo an no es abou he nume ic model & limi a ions
• This is a oy model (scala / enso p oxy). The eal enso Regge–Wheele /Ze illi
sys em has gauge sub le ies and cons ain equa ions; he code demons a es he
physics mechanism (exci a ion →QNMs + adia ion + s a ic/bound ail) a he han
p o iding p oduc ion- eady, gauge-in a ian wa e o ms o LIGO da a analysis.
• Bounda y condi ions a e simple app oxima e abso bing condi ions; o quan i a-
i e ingdown/echo p edic ion, you’d eplace hose wi h imp o ed ou going/ingoing
cha ac e is ic ea men s o ex end he domain.
• The “Hawking empe a u e co ec ion” p in ed is a quick es ima e using he Yukawa
ϵ( )ansa z; wi h he oy pa ame e s I used, i p oduced a la ge ac ional shi ( his
is because I used la ge p oxy ampli udes o demons a ion). Fo physical scena -
ios, you mus use ealis ic, cons ained ϵH≪1(Planck-supp essed), which gi es
iny ac ional shi s as discussed in ea lie sec ions.
6
• Nume ical dissipa ion and ini e domain make he la e- ime s a ic p o ile small; o
con e ge he Yukawa p o ile mo e closely, you can inc ease domain size, esolu-
ion, and use highe -o de disc e iza ion / be e bounda y condi ions.
7
Residual Space ime De o ma ions (RSD) and
Holog aphic P inciples
Oc obe 5, 2025
1 RSD s. he Holog aphic P inciple and So -Hai
P oposals
1.1 How Bounda y Da a Encodes Bulk Me ic De o ma ions (AdS
In ui ion)
In he AdS/CFT co espondence, he Fe e man–G aham expansion p o ides a
p ecise mapping be ween asymp o ic me ic coe icien s and bounda y da a. In
(d+ 1) bulk dimensions, nea he con o mal bounda y (coo dina e z→0):
ds2=dz2
z2+1
z2(g(0)ij(x) + z2g(2)ij(x) + ···+zdg(d)ij(x) + ···)dxidxj.(1)
•g(0)ij is he bounda y me ic and se es as a sou ce o he bounda y s ess
enso Tij.
•g(d)ij ( he no malizable mode) con ains he esponse/expec a ion alue hTiji.
I a esidual space ime de o ma ion (RSD) ield ∆µν has a nonze o bounda y
componen δg(0)ij o shi s no malizable pa s, i is encoded in bounda y da a
and, he e o e, in AdS, ully cap u ed by he dual CFT s a e. Conc e ely:
∆⇐⇒ δg(0)ij (sou ce) o δg(d)ij ( esponse) ⇐⇒ ope a o inse ion / s a e change in CFT.
(2)
Implica ion. I RSD p oduces an asymp o ic de o ma ion ha is ei he non-
no malizable (sou ce-like) o changes he no malizable mode, a holog aphic dual
will eco d ha in o ma ion in bounda y da a — consis en wi h holog aphy’s
bookkeeping o bulk deg ees o eedom. In AdS, he mapping is p ecise; in
asymp o ically- la space, he map is mo e sub le, bu so modes p o ide an
analogous bookkeeping channel (see nex subsec ion).
1
4.4 Obse a ional Cons ain s
•Sola -sys em / bina y pulsa es s: Residual long- ange ∆mus be negligi-
ble a hose scales. This implies ei he mla ge enough so ∆is sho - ange,
o ac i a ion unc ion A(K)ex emely localized. Quan i a i e cons ain :
he pos -New onian pa ame e de ia ions om GR a e measu ed a ≲10−5–
10−6in many egimes; ∆mus induce co ec ions smalle han hose.
•G a i a ional-wa e da a: Echo ampli udes |R|a e cons ained by LIGO/Vi go
analyses a he pe cen –subpe cen le el o loud signals. Use ou echo am-
pli ude o mulas o ansla e null de ec ions in o uppe bounds on ϵHand
m.
4.5 Mic ophysical O igin Requi ed
The EFT ac ion is agnos ic abou mic ophysics. To be ully con incing, one mus
p opose a mic ophysical o igin (s ingy condensa e, loop-quan um-g a i y em-
nan , en anglemen s uc u e change). Un il such a de i a ion is a ailable, RSD
emains an e ec i e hypo hesis. Howe e , he EFT is sel -consis en a linea
o de and p edic s es able signa u es.
5 Conc e e Nex S eps — Calcula ions & Checks o
Make RSD Fully Robus
1. Nonlinea s abili y analysis. Expand ac ion o cubic o de in ∆and check
o ghos s/ unaway solu ions. I p esen , a emp a dRGT-like comple ion o
show supp ession scale is Planckian.
2. Canonical quan iza ion o ∆in Schwa zschild backg ound: compu e mode
unc ions, canonical inne p oduc , and p ojec sou ce Jµν on o modes o
ge αk. Tha gi es explici S∆.
3. En opy bookkeeping. Cons uc mic os a e coun ing model o memo y
deg ees (e.g., quan ize ini e numbe No low-lying modes wi h cu o a
ho izon Planck-scale) and e i y whe he ∑kSkcan plausibly each A/4G
o only a subleading ac ion.
4. Nume ical ela i i y wi h ∆.Implemen he modi ied equa ions in a 3+1
code (wi h cons ain en o cemen ∇µ∆µν = 0) o simula e collapse and
measu e Qℓm, ene gy ans e , and inal ∆.
5. Obse a ional pipelines. P oduce empla e banks o echoes om RSD
laye s, and compu e ma ched- il e SNR agains LIGO da a o place bounds
on ϵHand m.
6. Mic ophysical de i a ion. Explo e candida e mic oscopic models (s ing
condensa es, spin-ne wo k ea angemen s) and compu e Jµν om i s p in-
ciples.
8
6 Sho Summa y / Takeaway
•RSD uni ies and ex ends so hai and g a i a ional memo y: he asymp-
o ic shea piece o RSD maps o so hai , while bulk Yukawa-like esiduals
ex end he s o age in o ini e and so can encode mo e de ailed mic os a e
in o ma ion.
•Ma hema ically consis en EFT: The co a ian ac ion + cons ain ensu es
compa ibili y wi h Bianchi iden i ies; he linea ized Lichne owicz dynam-
ics explains how ansien high-cu a u e sou ces can lea e pe manen ho-
mogeneous pieces.
•Key challenges: Ghos - ee non-linea comple ion, mic ophysical de i a-
ion, and obse a ional cons ain s (mus be small in weak ield bu possi-
bly obse able in ex eme e en s like me ge s).
•P omising es s: G a i a ional-wa e echoes, iny Hawking-spec um dis-
o ions (analogue BHs o hypo he ical p imo dial e apo a ing holes), and
ca e ul asymp o ic lux/so -cha ge bookkeeping.
A De ailed Tenso De i a ions
A.1 Con en ions and S a ing Iden i ies
We use signa u e (−+ ++),∇ he Le i–Ci i a connec ion o gµν , and geome ic
uni s (G=¯h=c= 1) unless no ed. Indices a e aised/lowe ed wi h g. Va ia ion
δdeno es unc ional de i a i e w. . . he a gumen indica ed.
Use ul iden i ies ( o a ia ions abou a backg ound ¯gµν):
δgµν =−gµαgνβδgαβ, δ√−g=−1
2√−ggµνδgµν.(24)
Va ia ion o Ch is o el symbols:
δΓα
βγ =1
2gαλ(∇βδgγλ +∇γδgβλ −∇λδgβγ).(25)
Va ia ion o Riemann:
δRα
βγδ =∇γδΓα
βδ −∇δδΓα
βγ.(26)
Con ac o ge he s anda d linea ized Ricci a ia ion:
δRµν =1
2(−□δgµν +∇µ∇νδg +∇µ∇αδgαν −∇ν∇αδgαµ),(27)
whe e δg ≡gαβδgαβ and □≡gαβ∇α∇β.
Finally,
δR =gµνδRµν −Rµνδgµν, δGµν =δRµν −1
2gµνδR −1
2δgµν R. (28)
9
A.2 Linea iza ion Abou a Backg ound and he Lichne owicz
Ope a o
Le gµν = ¯gµν +hµν wi h |h| 1. De ine he ace h≡¯gµν hµν and he ace- e e sed
pe u ba ion ¯
hµν =hµν −1
2¯gµνh.
Compu e δRµν wi h δgµν =hµν. Using he o mula abo e and eo ganizing
(s anda d manipula ion: mo e de i a i es, use commu a o s o co a ian de i a-
i es exp essed ia backg ound Riemann), one ob ains:
δRµν =−1
2□hµν −¯
Rα β
µ ν hαβ +∇(µ∇αhν)α−1
2∇µ∇νh, (29)
whe e all objec s a e wi h espec o ¯g. The linea ized Eins ein enso is:
δGµν =δRµν −1
2¯gµνδR
=1
2[−□hµν −2¯
Rα β
µ ν hαβ + 2 ¯
R(µαhν)α+∇µ∇νh−2∇(µ∇αhν)α
−¯gµν(∇α∇βhαβ −□h)].
(30)
In ha monic (de Donde ) gauge de ined by:
χν≡ ∇µ¯
hµν = 0,(31)
hese e ms simpli y s ongly, and one inds he compac ope a o o m:
δGµν =1
2∆Lhµν ,(32)
whe e ∆Lis he Lichne owicz ope a o ac ing on symme ic 2- enso s:
(∆Lh)µν ≡ −□hµν + 2 ¯
Rα β
µ ν hαβ −2¯
R(µαhν)α.(33)
Rema ks / de i a ion s eps: When commu ing co a ian de i a i es in e ms
such as ∇µ∇αhαν, you use [∇µ,∇α] β=Rµαβγ γ, p oduce he ¯
Rµανβhαβ e m, and
he backg ound Ricci e m ¯
Rµν yields he las e m abo e.
This is he linea wa e ope a o con olling me ic pe u ba ions; i is he
p incipal ope a o used h oughou he pape .
A.3 Eule –Lag ange Equa ion o S∆and he ∆-EOM
Take he ac ion (minimal model) used elsewhe e:
S∆=1
32π∫d4x√−g[∆µν(L∆)µν +m2(∆µν∆µν −α∆2)],(34)
wi h ∆≡gµν∆µν . Fo cla i y, we will se α= 1 (Fie z–Pauli uning).
Va y S∆wi h espec o he independen ield ∆ρσ. Keeping he me ic ixed
o his a ia ion and in eg a ing by pa s o make he ope a o symme ic (d op
bounda y e ms), we ge he Eule –Lag ange equa ion:
1
32π[(L∆)ρσ −m2(∆ρσ −gρσ∆)]=−δSac
δ∆ρσ .(35)
10
I he ac i a ion coupling is Sac =∫√−gA(K)Jµν∆µν +∫√−gλν∇µ∆µν, hen:
δSac
δ∆ρσ =√−g(A(K)Jρσ −∇(ρλσ)).(36)
Collec ing ac o s and mul iplying bo h sides by 32πyields he co a ian o m
used in he main ex :
(L∆)µν −m2(∆µν −gµν∆) = −16πA(K)Jµν + 2∇(µλν).(37)
Va ying w. . . λνen o ces he cons ain :
∇µ∆µν = 0.(38)
This is an explici de i a ion o he mas e PDE ( he ac o con en ions ma ch
he ac ion no maliza ion).
A.4 Elimina ing λand Di e gence Cons ain
Take he co a ian di e gence ∇µo he EOM:
∇µ(L∆)µν −m2∇µ(∆µν −gµν∆) = −16π∇µ(AJµν)+2∇µ∇(µλν).(39)
Use ∇µ∆µν = 0 and cu a u e-de i a i e iden i ies o sol e o λν. In many
analyses, one chooses ini ial da a so ha λν= 0 (o i can be abso bed in o a
gauge ede ini ion), lea ing he simpli ied o m:
(L∆)µν −m2(∆µν −gµν∆) = −16πAJµν.(40)
This is he wo king mas e equa ion used in he pape .
A.5 S ess–Ene gy Tenso T∆
µν om S∆(Fo mal Expansion)
De ine:
T∆
µν ≡ − 2
√−g
δS∆
δgµν .(41)
Va ying S∆w. . . he me ic p oduces se e al ypes o e ms:
1. Explici me ic ac o s ( om √−gand index aising in ∆µν).
2. Va ia ions o he ope a o L(because i con ains cu a u e and connec ion).
3. Va ia ions o he mass e m.
Collec ing and a anging (leng hy bu s aigh o wa d) gi es he schema ic
bu explici s uc u e:
T∆
µν =1
16π{−1
2gµν[∆αβ(L∆)αβ −m2(∆αβ∆αβ −∆2)]
+ ∆(µα(L∆)ν)α+ (symme ized de i a i e e ms)
−m2(2∆µα∆να−gµν (∆αβ∆αβ −∆2))+Cµν[∆; R]}.
(42)
11
He e, Cµν[∆; R]deno es e ms ha come om a ying cu a u e enso s in-
side Land so a e linea in backg ound cu a u e imes quad a ic in ∆(schema -
ically o he o m R∆2o ∆∇∇∆a e in eg a ions by pa s). W i ing he ully
index-expanded exp ession is unwieldy bu s aigh o wa d; he boxed s uc-
u e shows he ypes o con ibu ions and he key on-shell simpli ica ion:
On shell (use EOM (L∆) −m2(···) = S), many e ms simpli y, and one may
exp ess pa s o T∆in e ms o he sou ce Sµν and di e gences he eo , which
p o es conse a ion ∇µT o
µν = 0 when ma e EOM hold.
B Compu a ional No es & Pe u ba i e Expansions
B.1 Linea Gauge, T ace Re e sal, and Cons ain s
Gauge ans o ma ions a linea o de a e gene a ed by a ec o ield ξµ:
hµν 7→ hµν +∇µξν+∇νξµ.(43)
T ace- e e sed ield ¯
hµν =hµν −1
2gµνhobeys:
¯
hµν 7→ ¯
hµν +∇µξν+∇νξµ−gµν∇αξα.(44)
Ha monic gauge ∇µ¯
hµν = 0 ixes esidual gauge up o Killing ec o s o he
backg ound. Fo nume ical e olu ion, one ypically imposes his gauge (o equi -
alen gene alized-ha monic condi ion) o ob ain a mani es ly hype bolic sys em.
Fo ou ∆µν ield, we en o ced ∇µ∆µν = 0; when sol ing nume ically, one mus
main ain his cons ain (p ojec o di e gence- ee subspace a each ime-s ep
o e ol e he Lag ange mul iplie ield).
B.2 Mode Decomposi ion on Schwa zschild — E en/Odd Pa -
i y & Mas e Equa ions
Because he backg ound is sphe ically symme ic, expand enso pe u ba ions
in enso sphe ical ha monics. Fo each (ℓ, m)mode, he e exis wo pa i y sec-
o s:
•Odd (axial) pa i y: Go e ned by he Regge–Wheele mas e a iable Ψodd
ℓm ,
which sa is ies: [−∂2
+∂2
∗−VRW
ℓ( )]Ψodd
ℓm =Sodd
ℓm ,(45)
wi h
VRW
ℓ( ) = ( )(ℓ(ℓ+ 1)
2−6M
3), ( ) = 1 −2M
.(46)
•E en (pola ) pa i y: Go e ned by he Ze illi mas e a iable Ψe en
ℓm , which
sa is ies: [−∂2
+∂2
∗−VZ
ℓ( )]Ψe en
ℓm =Se en
ℓm .(47)
12
De ini ion: λ≡1
2(ℓ−1)(ℓ+ 2). A compac o m o he Ze illi po en ial is
(s anda d ex book o m):
VZ
ℓ( ) = 2
3
λ2(λ+ 1) 3+ 3λ2M 2+ 9λM2 + 9M3
(λ + 3M)2.(48)
(One can expand and simpli y; he abo e is he con en ional compac exp es-
sion wi h λsho hand.)
Mapping o ∆µν.The componen s o ∆µν a e combined ( ia algeb aic ela-
ions de i ed om he linea ized ield equa ions and he gauge cons ain ) in o
he mas e a iables Ψ(odd/e en)
ℓm . The same educ ion applies o he ∆- ield be-
cause each enso ha monic componen sa is ies a wa e-like equa ion wi h an
e ec i e po en ial plus a mass e m (−m2Ψ) and a p ojec ed sou ce Sℓm buil om
AJµν.
B.3 Nume ical Disc e iza ion and S abili y (P ac ical Recipe)
Coo dina e choice. Wo k in o oise coo dina e ∗de ined by d ∗/d = 1/ ( ).
Use a uni o m g id in ∗ o a oid Cou an mapping issues om a iable wa e
speeds in .
Fini e-di e ence scheme (2nd-o de accu a e leap og).
Le Ψn
ideno e Ψ( n, ∗i). The upda e o mula o he 1D wa e equa ion:
∂2
Ψ−∂2
∗Ψ + V( )Ψ + m2Ψ = S( , )(49)
is (cen e ed ime & space):
Ψn+1
i= 2Ψn
i−Ψn−1
i+ ∆ 2(Ψn
i+1 −2Ψn
i+ Ψn
i−1
∆ 2
∗−(Vi+m2)Ψn
i+Sn
i).(50)
CFL condi ion (s abili y): Fo 1D wa e speed c= 1 in ∗, equi e:
∆ ≤∆ ∗.(51)
In p ac ice, ake ∆ =C∆ ∗wi h C≲0.5 o s abili y and accu acy.
Bounda y condi ions.
•Ou e bounda y (la ge posi i e ∗): Apply Somme eld/ou going: ∂ Ψ +
∂ ∗Ψ = 0. Disc e ize as Ψn+1
N= Ψn
N−1+ (1 −∆ /∆ ∗)(Ψn
N−Ψn
N−1)(p ac ical
one-sided o mula) o use sponge/abso bing laye s.
•Inne bounda y (nea ho izon ∗→ −∞): Impose ingoing bounda y ∂ Ψ−
∂ ∗Ψ=0o place inne bounda y su icien ly inside he ho izon and use
egula i y.
Cons ain en o cemen o ∇µ∆µν = 0.Ei he :
• Decompose on o di e gence- ee enso ha monics so modes a e au oma -
ically di e gence- ee; o
• E ol e he Lag ange mul iplie λν(coupled equa ions) and impose con-
s ain dampe s (Gundlach–Husa s yle cons ain damping) o p e en g ow h
o iola ions.
13
B.4 G een’s Func ion App oach and Low-F equency Expansion
Fo linea ope a o D=−∂2
+∂2
∗−V( )−m2, he e a ded G een’s unc ion sol es:
DxG e (x, x′) = δ(2)(x−x′).(52)
Spec al decomposi ion ( o disc e e + con inuous spec um) yields he usual
ep esen a ion:
G e ( , ∗; ′, ′
∗) = Θ( − ′){∑
n
ψn( ∗)ψn( ′
∗)
ωn
sin[ωn( − ′)]+∫∞
ωmin
dωψω( ∗)ψω( ′
∗)
ωsin[ω( − ′)]}.
(53)
When m > 0, he con inuum begins a ωmin =m; he s a ic piece co esponds
o ω= 0 and exis s only i he e a e ze o- equency (bound) modes o as he
ω→0limi o he con inuum when m= 0.
S a ic ( ime-independen ) G een unc ion. Sol e:
(∂2
∗−V( )−m2)Gs a ( ∗, ′
∗) = −δ( ∗− ′
∗).(54)
Then, o a compac -in- ime sou ce S( , ) = q( ) ( ), he la e- ime ( → ∞)
ield con ains he con ibu ion:
Ψs a ( ) = [∫∞
−∞
( )d ]·∫d ′
∗Gs a ( ∗, ′
∗)q( ′
∗).(55)
This is he igo ous s a emen behind he Yukawa s a ic ail: in la space,
Gs a (x,x′) = e−m|x−x′|
4π|x−x′|.
C Connec ion o E ec i e Field Theo y (EFT)
C.1 Field Con en , Symme ies, and Ope a o Expansion
T ea gµν and he symme ic enso ∆µν as ields in an e ec i e low-ene gy La-
g angian. The mos gene al di eomo phism-in a ian local ac ion (up o second
o de in de i a i es) is:
Se =1
16πG ∫d4x√−gR +∫d4x√−g[L∆+Lin +∑
n≥1
cn
ΛnOn+4],(56)
whe e L∆con ains he kine ic ( wo-de i a i e) and mass-like e ms o ∆:
L∆∼1
32π(∆L∆−m2(∆µν∆µν −∆2)),(57)
Lin con ains allowed couplings o cu a u e and ma e (e.g., ∆R,∆T,∆R∆),
and On+4 a e highe -dimension ope a o s supp essed by he EFT cu o Λ(ex-
pec ed ∼MPl unless new physics lowe s i ).
Symme y cons ain s.
• Di eomo phism in a iance es ic s ope a o s o be scala s buil om g,
∆, and co a ian de i a i es.
14
• I ∆ ans o ms as a enso unde di eos, he in e ac ion e ms mus e-
spec ha ; i we wan o allow only a ans e se ∆sec o , we may impose
( h ough λ)∇µ∆µν = 0 as an ope a o -le el cons ain .
C.2 Powe Coun ing and Na u al Sizes
Ope a o dimensions and na u al sizes:
• The wo-de i a i e kine ic e m se s no maliza ion o ∆. I ∆is no malized
simila ly o linea ized me ic pe u ba ions, one can assign he same mass
dimension as a canonically no malized spin-2 ield in 4D (dimension 1).
Then, he mass e m has dimension m(mass pa ame e ).
• Highe -o de ope a o s (e.g., (∇∆)2∆/Λ,R∆2/Λ,∆4/Λ0wi h coe icien s sup-
p essed by powe s o Λ) a e na u al loop-gene a ed e ms. Loops o ∆and
ma e gene a e eno maliza ion o hese coe icien s; gene ically, one ex-
pec s cn∼O(1) unless p o ec ed by symme y.
Cu o and egime o alidi y. The EFT is alid o cu a u es K Λ4, mo-
men a kΛ. I we ake Λ∼MPl, hen he EFT applies well below Planckian
cu a u e, bu ac i a ion A(K)in ou model in en ionally u ns on nea Planck-
scale cu a u e — his equi es cau ion: ei he ea he heo y as an e ec i e
pa ame e iza ion o unknown UV physics (accep able, bu a oid s ong claims
abou UV comple ion), o embed i in a con olled UV model.
C.3 Fie z–Pauli Tuning and Ghos s; Nonlinea Comple ion
A linea o de , a gene ic mass e m o a spin-2 ield in oduces a ghos (Boulwa e–
Dese ). The Fie z–Pauli mass s uc u e:
Lmass ∝m2(∆µν∆µν −∆2)(58)
emo es he ex a scala ghos a linea o de . Nonlinea ly, a oiding he BD
ghos gene ally equi es a special s uc u e (dRGT massi e g a i y) — cons uc -
ing a ull ghos - ee non-linea comple ion o ∆while keeping gµν dynamical is
non i ial.
P ac ical EFT app oach used in he pape :
• Wo k pe u ba i ely and assume ∆is small ( alid i ac i a ion is localized
and Planck-supp essed a mac oscopic scales). In his egime, linea s abil-
i y and absence o ghos modes su ice o con olled p edic ions.
• Al e na i ely, in e p e ∆as a non-p opaga ing condensa e (an expec a ion
alue o mic ophysics) a he han a new undamen al p opaga ing spin-2
— hen he ghos issue is less se e e.
15
C.4 Reno maliza ion-G oup (RG) Rema ks and Loop Gene a-
ion
Loops o ma e ields couple o ∆ ia Jµν and will eno malize coe icien s in L∆
(including gene a ing e ms like R∆2,(∇∆)2, e c.). Powe -coun ing es ima e:
• A loop wi h cu o Λwill ypically p oduce co ec ions δm2∼κΛ2and δcn∼
κ(dimensionless), wi h κa loop ac o . Thus, keeping msmall compa ed
o Λ equi es uning o symme y p o ec ion. This is he usual na u alness
ension common o many EFTs.
Sensible model-building choices:
• Take Λ∼MPl and mo o de Planck — hen ∆is sho - ange, and e ec s on
as ophysical scales a e exponen ially small.
• Al e na i ely, ake msmall bu assume UV comple ion supplies a symme y
o mechanism p o ec ing m(e.g., pa ially massless limi s, gauge symme y
enhancemen , o dynamical sc eening like Vainsh ein mechanism).
C.5 Ma ching o Mic ophysics (Ske ch)
I one had an explici UV heo y (s ing compac i ica ion, spin-ne wo k conden-
sa es, e c.), in eg a e ou hea y deg ees o ind he e ec i e couplings:
A(K)Jµν ←− hOµνiUV (59)
whe e Oµν is some composi e ope a o o he UV heo y ha becomes a local
enso sou ce in he IR. The shape o A( h eshold ac i a ion s. smoo h) is se
by he de ails o he spec al densi y o UV luc ua ions nea Kc i .
16
De i a ion: F om a Thin Null Shell (Vaidya) o
Ha monic Sou ce Coe icien s
Oc obe 5, 2025
1 De i a ion: F om a Thin Null Shell (Vaidya) o Ha -
monic Sou ce Coe icien s
1.1 Se up (Ingoing Vaidya / Thin Null Shell)
Use ingoing Edding on–Finkels ein (EF) coo dina es ( , , θ, φ)wi h me ic:
ds2=−(1−2M( , θ, φ)
)d 2+ 2d d + 2dΩ2.(1)
Fo ingoing null dus , he s ess-ene gy is (exac ly):
Tµν =˙
M( , θ, φ)
4π 2ℓµℓν, ℓµ=−∂µ , (2)
whe e ˙
M≡∂ Mand he only nonze o coo dina e componen in EF coo di-
na es is:
T =˙
M( , θ, φ)
4π 2.(3)
1.2 Angula Decomposi ion
Expand he (angle-dependen ) mass unc ion in sphe ical ha monics:
M( , θ, φ) = ∑
ℓ,m
Mℓm( )Yℓm(θ, φ).(4)
Di e en ia e in ime:
˙
M( , θ, φ) = ∑
ℓ,m
˙
Mℓm( )Yℓm(θ, φ).(5)
1
I execu ed he code and sa ed plo s o /mn /da a/:
•/mn /da a/exac shellwa e o m.png—obse e wa e o ma obs = 120M.
•/mn /da a/exac shellI .png—in eg a edp oxyI( ) = ∫Ψ2d .
•/mn /da a/exac shellPsi inal.png—la e − ime adialp o ile.
(Those iles we e jus w i en by he no ebook — you can download hem
om he en i onmen .)
8 Quick In e p e a ion o he Nume ic Run
• The wa e o m a la ge adius shows he expec ed p omp exci a ion ( he
shell injec ion), a ai ly clean quasino mal-mode-like ingdown, and la e
small bumps consis en wi h pa ial e lec ions/echo-like ea u es (same
quali a i e beha iou as be o e).
• The in eg a ed in o ma ion p oxy I( )spikes du ing he shell and se les o
a small bu nonze o la e- ime pla eau, consis en wi h a esidual s o ed in
he mas e ield ( his is p ecisely he RSD / geome ic memo y e ec you a e
s udying).
• The la e- ime adial p o ile exhibi s oscilla o y ail beha io and a nonze o
nea -sou ce ampli ude (again consis en wi h a localized esidual sou ced
by he shell).
Nume ically, e e y hing is s able and beha es as expec ed o he chosen pa-
ame e s. The ampli ude depends s ongly on he shell ampli ude Aℓm and he
chosen egula iza ion wid h σ .
8
